Answers to Odd-Numbered Exercises for. A Visual Approach to SPSS for.
Windows. Leonard Stern. Eastern Washington University ...
Answers to Odd-Numbered Exercises
for
A Visual Approach to SPSS for Windows
Leonard Stern Eastern Washington University
TABLE OF CONTENTS INTRODUCTION ......................................................................................................................................... I CHAPTER 2 ................................................................................................................................................. 2 Exercise 2.1........................................................................................................................................... 2 Exercise 2.3........................................................................................................................................... 2 Exercise 2.5........................................................................................................................................... 3 Exercise 2.7........................................................................................................................................... 5 CHAPTER 3 ............................................................................................................................................... 10 Exercise 3.1......................................................................................................................................... 10 CHAPTER 4 ............................................................................................................................................... 12 Exercise 4.1......................................................................................................................................... 12 Exercise 4.3......................................................................................................................................... 13 CHAPTER 5 ............................................................................................................................................... 14 Exercise 5.1......................................................................................................................................... 14 Exercise 5.3......................................................................................................................................... 17 CHAPTER 6 ............................................................................................................................................... 20 Exercise 6.1......................................................................................................................................... 20 Exercise 6.3a....................................................................................................................................... 22 Exercise 6.3b....................................................................................................................................... 25 Exercise 6.5......................................................................................................................................... 29 CHAPTER 7 ............................................................................................................................................... 32 Exercise 7.1......................................................................................................................................... 32 Exercise 7.3......................................................................................................................................... 34 CHAPTER 8 ............................................................................................................................................... 39 Exercise 8.1......................................................................................................................................... 39 Exercise 8.3......................................................................................................................................... 42 CHAPTER 9 ............................................................................................................................................... 44 Exercise 9.1......................................................................................................................................... 44 Exercise 9.3......................................................................................................................................... 48 CHAPTER 10 ............................................................................................................................................. 51 Exercise 10.1....................................................................................................................................... 51 Exercise 10.3....................................................................................................................................... 55 Exercise 10.5....................................................................................................................................... 59 CHAPTER 11 ............................................................................................................................................. 60 Exercise 11.1....................................................................................................................................... 60 Exercise 11.3....................................................................................................................................... 64 Exercise 11.5....................................................................................................................................... 65 CHAPTER 12 ............................................................................................................................................. 67 Exercise 12.1....................................................................................................................................... 67 Exercise 12.3....................................................................................................................................... 68 Exercise 12.5....................................................................................................................................... 70 CHAPTER 13 ............................................................................................................................................. 72
Exercise 13.1....................................................................................................................................... 72 Exercise 13.3....................................................................................................................................... 74 Exercise 13.5....................................................................................................................................... 76 CHAPTER 14 ............................................................................................................................................. 79 Exercise 14.1....................................................................................................................................... 79 Exercise 14.3....................................................................................................................................... 85 CHAPTER 15 ............................................................................................................................................. 89 Exercise 15.1....................................................................................................................................... 89 Exercise 15.3....................................................................................................................................... 93 CHAPTER 16 ............................................................................................................................................. 94 Exercise 16.1....................................................................................................................................... 94 Exercise 16.3....................................................................................................................................... 98 CHAPTER 17 ........................................................................................................................................... 104 Exercise 17.1..................................................................................................................................... 104 Exercise 17.3..................................................................................................................................... 105 Exercise 17.5..................................................................................................................................... 106 Exercise 17.7..................................................................................................................................... 108 CHAPTER 18 ........................................................................................................................................... 110 Exercise 18.1..................................................................................................................................... 110 Exercise 18.3..................................................................................................................................... 117 CHAPTER 19 ........................................................................................................................................... 125 Exercise 19.1..................................................................................................................................... 125 Exercise 19.3..................................................................................................................................... 130
Introduction This supplement provides answers to the odd-numbered exercises presented at the end of each chapter. Because some exercises require that tasks be completed (rather than answers given), the material provided here sometimes demonstrates how the tasks are to be accomplished and may provide all or some of the resulting output produced. The answers shown here often include a screen shot of the Data View of the SPSS Statistics Data Editor and may be preceded by a display of the screens required to produce the data. Exercises that call for exploration such as the exercises at the end of Chapters 1 and 2 are not answered.
i
Chapter 2 Exercise 2.1 The data are type in using the keyboard to look like this:
Exercise 2.3 The data file is opened starting from the File menu:
1. From the File menu select Open and then Data.
The drop-down menu is used to select Excel as the file type. The desired file can be opened by double-clicking it.
2
The file is read into SPSS by clicking OK in the window that opens:
The file is then read in:
Exercise 2.5 A blank file is opened from the File menu by selecting New and then Data:
3
The data are typed in using the keyboard. The finished file is shown below:
The file is saved using the name EX25.sav.
The file name is entered here.
4
Exercise 2.7 The file is opened from the File menu by selecting Open and then Data:
A .dat file is selected as the file type:
5
The file name is entered here.
The Text Import Wizard is used to read in the file:
6
7
8
9
Chapter 3 Exercise 3.1 1.
The data are typed in using the keyboard to look like this:
2.
The new variable is created by highlighting column 2 and choosing Insert Variable from the Edit menu.
3.
The variable is deleted by highlighting the column and choosing Cut from the Edit menu.
10
4.
Values of the new variable are type in from the keyboard.
5.
The file is saved using Save As from the File menu.
11
Chapter 4 Exercise 4.1 1.
The file looks like this:
The file’s Variable View looks like this:
12
Exercise 4.3 The file contains 20 cases. Its Data View looks like this:
The file’s Variable View looks like this:
13
Chapter 5 Exercise 5.1 1.
The file is opened using the File menu and selecting Open and then Data.
2.
Select Descriptive Statistics then Frequencies from the Analyze menu and run the Frequencies procedure.
Select the variable tvhrs and transfer it to the Variables pane
Click OK.
14
3.
Replace the word Frequencies in the title with the new title information.
4.
In the outline pane, click and drag the icon for the table to the icon for Title.
15
5.
Click on the table to be copied then select Copy from the Edit menu.
Paste the table into the desired word-processing document.
16
Exercise 5.3 The file can be opened using the File menu in SPSS and selecting Open and then Output. Then the Export procedure is selected from the File menu.
17
Select All Verify this option is selected.
Insert the proper file name here.
18
Note that the wide table has been shrunk to fit the page.
19
Chapter 6 Exercise 6.1
1. From the Graphs menu choose the Chart Builder.
2. Using the Chart Builder select Line as the chart type.
3. Drag the icon of the multiple line plot from here to here.
20
4. Drag the nominal scale condition to the x-axis drop zone.
5. Drag the nominal scale gender to the grouping drop 6. Drag the scale variable dv to the y-axis drop zone.
7. Click OK.
21
Exercise 6.3a
1. From the Graphs menu choose the Chart Builder.
22
2. From the Gallery tab select Bar.
3. Drag the icon of the bar chart from here to the chart canvas.
4. Drag the nominal scale highest to the x-axis drop zone.
23
5. Press the Titles/Footnotes tab and select Title 1.
6. Type the title in the Content field of the Element Properties window.
7. Click Apply then OK.
24
Exercise 6.3b
1. From the Graphs menu choose the Chart Builder.
25
2. From the Gallery tab select Bar.
3. Drag the icon of the clustered bar chart from here to the chart canvas.
4. Drag the nominal scale highest to the x-axis drop zone. 5. Drag the nominal scale gender to the grouping drop zone.
26
6. From the Statistic drop-down menu of the Element Properties window select Percentage (). 7. Click Set Parameters.
8. Select Total for Each Legend Variable Category (same fill color).
9. Click Continue.
10. Click Apply.
11. Press the Titles/Footnotes tab and select Title 1.
27
12. Type the title in the Content field of the Element Properties window.
13. Click Apply then OK.
28
Exercise 6.5
1. From the Graphs menu choose the Chart Builder.
2. From the Gallery tab select Scatter/Dot.
3. Drag the icon of the simple scatterplot from here to the chart canvas.
29
4. Drag homework1 to the x-axis drop zone.
5. Drag quiz1 to the y-axis drop zone.
6. Click the Titles/Footnotes tab and select Title 1.
30
7. Type the title in the Content field of the Element Properties window.
13. Click Apply then OK on the Chart Builder window shown on the previous page.
31
Chapter 7 Exercise 7.1 1.
The Recode into Different Variables procedure provides the requested transformations:
32
2.
The Compute Variable expression shown below creates the variable newsfreq1.
The variables created in problems 1 and 2 are shown below: 33
Exercise 7.3 1.
The variable seqnum is set to the value of $Casenum that is obtained from the miscellaneous function group.
34
2.
The variable nocontact can be created in two steps. First, all cases can be declared to have the value of nocontact be 0:
Then, using the IF button, nocontact can be set to 1 when two other conditions are met
35
Press this button to form the If statement.
Select this option.
3.
The desired selections are made using the Select Cases procedure with an IF statement: 36
This statement appears after the window below has been filled out.
37
These 2 cases are selected (as well as case 347).
38
Chapter 8 Exercise 8.1 1.
The Frequencies procedure provides the requested output:
39
The output is shown next:
Statistics Body temperature (degrees Fahrenheit) N
Valid
130
Missing
0 98.249 .7332 -.004 .212 .780 .422 97.800
Mean Std. Deviation Skewness Std. Error of Skewness Kurtosis Std. Error of Kurtosis Percentiles 25 50
98.300
75
98.700
40
Body temperature (degrees Fahrenheit) Frequency Valid
Percent
96.3
1
96.4
1
96.7
2
96.8
1
96.9
Valid Percent .8
Cumulative Percent
.8
.8
.8
.8
1.5
1.5
1.5
3.1
.8
.8
3.8
1
.8
.8
4.6
97.0
1
.8
.8
5.4
97.1
3
2.3
2.3
7.7
97.2
3
2.3
2.3
10.0
97.3
1
.8
.8
10.8
97.4
5
3.8
3.8
14.6
97.5
2
1.5
1.5
16.2
97.6
4
3.1
3.1
19.2
97.7
3
2.3
2.3
21.5
97.8
7
5.4
5.4
26.9
97.9
5
3.8
3.8
30.8
98.0
11
8.5
8.5
39.2
98.1
3
2.3
2.3
41.5
98.2
10
7.7
7.7
49.2
98.3
5
3.8
3.8
53.1
98.4
9
6.9
6.9
60.0
98.5
3
2.3
2.3
62.3
98.6
10
7.7
7.7
70.0
98.7
8
6.2
6.2
76.2
98.8
10
7.7
7.7
83.8
98.9
2
1.5
1.5
85.4
99.0
5
3.8
3.8
89.2
99.1
3
2.3
2.3
91.5
99.2
3
2.3
2.3
93.8
99.3
2
1.5
1.5
95.4
99.4
2
1.5
1.5
96.9
99.5
1
.8
.8
97.7
99.9
1
.8
.8
98.5
100.0
1
.8
.8
99.2
100.8
1
.8
.8
100.0
Total
130
100.0
100.0
41
2.
The percentile rank of the temperature 98.6 is 70.
3.
Examination of the histogram and comparison of the index of skewness and kurtosis with each measure’s standard error indicates that, although there are too many scores in the center of the distribution, the distribution is approximately normal.
Exercise 8.3 1.
The Frequencies:Statistics window is used to request the table:
The Frequencies: Format window is completed as shown below: Selecting this option puts the variables in the same table.
42
Statistics MeanTime2003 N
Valid
MeanTime2001
65
65
Mean Std. Deviation Percentiles 25
35 23.328 4.0805 20.650
35 23.551 3.9571 21.100
35 23.54 3.899 21.05
50
22.600
23.300
22.80
75
25.550
25.300
25.65
Missing
2.
MeanTime2002
65
Mean commute times do not seem to have appreciably changed over the 3-year period.
43
Chapter 9 Exercise 9.1 1.
The requested output can be obtained using the Explore procedure:
The output is shown below: Case Processing Summary Cases Valid N Annual Rate of Change
Missing Percent
16
N
100.0%
Total
Percent 0
.0%
N
Percent 16
100.0%
Descriptives Statistic Annual Rate of Change
Mean
-.0089
95% Confidence Interval for Mean
Lower Bound
-.0495
Upper Bound
.0317
5% Trimmed Mean
-.0121
Median
-.0136
Variance
Std. Error .01904
.006
Std. Deviation
.07618
Minimum
-.14
Maximum
.18
Range
.32
Interquartile Range
.09
Skewness Kurtosis
44
.606
.564
1.453
1.091
45
2.
The shape of the distribution, although not perfectly normal, does not deviate significantly from normal according to measures of skewness and kurtosis (the z value for each measure, obtained by dividing its value by its standard error, is close to 1). The mean change is -.01 and the standard deviation is .08.
3.
The year with the highest rate of change in water usage was 1994. The data point represents an outlier, one that falls from 1.5 to 3 units of interquartile range beyond the upper edge of the box.
4.
Based on the output for the variable usageperc that is shown below, the distribution can be described as being approximately normal, having a mean of 114,841.30, and a standard deviation of 8,685.66. Descriptives Statistic
Water Usage per Capita
Mean
Std. Error
114841.3394 2171.41593
95% Confidence Interval for Mean
Lower Bound
110213.0759
Upper Bound
119469.6029
5% Trimmed Mean
114805.6318
Median
113709.9997
Variance
7.544E7
Std. Deviation
8685.66373
Minimum
99311.40
Maximum
131014.01
Range
31702.61
Interquartile Range
12810.62
Skewness Kurtosis
46
.265
.564
-.347
1.091
47
Exercise 9.3 1.
Z-scores are requested using the Descriptives procedure:
2.
Histograms can be requested from the Frequencies procedure.
48
49
The distributions are identical in shape.
50
Chapter 10 Exercise 10.1 1.
The scatterplot of car weight as a function of wheelbase is shown below:
2.
The output of the regression analysis is shown below: Descriptive Statistics Mean Weight WheelBase
Std. Deviation
3581.2146 108.1722
N
759.98462 8.34544
424 424
Correlations Weight Pearson Correlation
Weight
1.000
.761
.761
1.000
.
.000
WheelBase
.000
.
Weight
424
424
WheelBase
424
424
WheelBase Sig. (1-tailed) N
WheelBase
Weight
51
Variables Entered/Removedb Variables Entered
Model
Variables Removed
Method
a
1
WheelBase
. Enter
a. All requested variables entered. b. Dependent Variable: Weight
\ Model Summaryb Model
R .761a
1
Adjusted R Square
R Square .579
Std. Error of the Estimate
.578
493.80747
a. Predictors: (Constant), WheelBase b. Dependent Variable: Weight
ANOVAb Model 1
Sum of Squares
df
Mean Square
Regression
1.414E8
1
1.414E8
Residual
1.029E8
422
243845.818
Total
2.443E8
423
F
Sig.
579.924
.000a
a. Predictors: (Constant), WheelBase b. Dependent Variable: Weight
Coefficientsa Unstandardized Coefficients Model 1
B (Constant) WheelBase
Std. Error
-3913.224
312.133
69.282
2.877
a. Dependent Variable: Weight
52
Standardized Coefficients t
Beta
.761
Sig.
-12.537
.000
24.082
.000
Casewise Diagnosticsa Case Number
Std. Residual
300 305 307 323 326 338 348 418
Weight
3.263 3.628 3.316 3.531 3.193 3.126 3.161 -3.069
Predicted Value
7190.00 6400.00 5969.00 5590.00 5423.00 5390.00 4576.00 4548.00
5578.4787 4608.5238 4331.3938 3846.4163 3846.4163 3846.4163 3015.0263 6063.4562
Residual 1611.52125 1791.47624 1637.60624 1743.58373 1576.58373 1543.58373 1560.97373 -1515.45624
a. Dependent Variable: Weight
Residuals Statisticsa Minimum Predicted Value Residual Std. Predicted Value Std. Residual
2252.9187 -1515.45630 -2.297 -3.069
Maximum 6063.4561 1791.47620 4.293 3.628
a. Dependent Variable: Weight
53
Mean 3581.2146 .00000 .000 .000
Std. Deviation 578.19311 493.22343 1.000 .999
N 424 424 424 424
54
Here are the answers to the questions: a. Based on the two scatterplots, the relation between car weight and length of wheelbase appears to be linear. b. The r value is .76. c. From the ANOVA table, the relationship is statistically significant, F (1,422) = 579.92, p < .001. d. The equation is Weight = 69.2 × Wheelbase − 3913.22 e. Based on the histogram showing the distribution of standardized residuals and the normal P-P plot, the assumption of normality appears to hole. The plot showing the standardized predicted values on the x-axis and the standardized residuals on the y-axis does not reveal serious threats to the assumption of homogeneity of residuals.
Exercise 10.3 1.
The Model Summary and ANOVA tables from the linear regression analysis provide an answer:
Model Summary Model 1
R .081a
Adjusted R Square
R Square .007
-.014
a. Predictors: (Constant), Average pupil/teacher ratio
55
Std. Error of the Estimate 75.34596
ANOVAb Model 1
Sum of Squares Regression
df
Mean Square
1811.030
1
1811.030
Residual
272496.650
48
5677.014
Total
274307.680
49
F
Sig. .319
.575a
a. Predictors: (Constant), Average pupil/teacher ratio b. Dependent Variable: Mean total SAT in 1994-1995
The relation between mean SAT score and mean pupil/teacher ratio was not significant, r = .08, F < 1. 2.
A new variable LOGCOST was created with the Compute Variable procedure.
From the Model Summary, ANOVA, and coefficients tables of a linear regression analysis, the necessary data are available to draw a conclusion about the relation. Model Summaryb Model 1
R .392a
Adjusted R Square
R Square .154
.136
a. Predictors: (Constant), LOGCOST b. Dependent Variable: Mean total SAT in 1994-1995
56
Std. Error of the Estimate 69.55055
ANOVAb Model 1
Sum of Squares Regression
df
Mean Square
42118.295
1
42118.295
Residual
232189.385
48
4837.279
Total
274307.680
49
a. Predictors: (Constant), LOGCOST
F
Sig. .005a
8.707
The negative sign shows the relation is an inverse one.
b. Dependent Variable: Mean total SAT in 1994-1995
Coefficientsa Unstandardized Coefficients Model 1
B
Std. Error
(Constant)
1202.433
80.754
LOGCOST
-310.851
105.346
Standardized Coefficients t
Beta
-.392
Sig.
14.890
.000
-2.951
.005
a. Dependent Variable: Mean total SAT in 1994-1995
There was in inverse relation between log cost and mean SAT, r = -.39, F (1, 48) = 8.71, p < .01. Thus, the more money states spent, the lower the mean SAT score was. The scatterplot below illustrates the relation:
57
Histograms of the distributions of residuals before and after the log transformation of cost show a slight improvement in the degree to which the residuals are normally distributed around the regression line. After log transform.
Before log transform.
3.
The Model Summary, ANOVA, and coefficients tables of a linear regression analysis allow a conclusion to be drawn about the relation between the variables. Model Summaryb Model 1
R
Adjusted R Square
R Square
.887a
.787
Std. Error of the Estimate
.783
34.89065
a. Predictors: (Constant), Percent of all eligible students taking the SAT in 1994-1995 b. Dependent Variable: Mean total SAT in 1994-1995 ANOVAb Model 1
Sum of Squares Regression
Mean Square 1
215874.533
58433.147
48
1217.357
274307.680
49
Residual Total
df
215874.533
F
Sig.
177.330
.000a
a. Predictors: (Constant), Percent of all eligible students taking the SAT in 1994-1995 b. Dependent Variable: Mean total SAT in 1994-1995
Coefficientsa Unstandardized Coefficients Model 1
B (Constant) Percent of all eligible students taking the SAT in 1994-1995
Std. Error
1053.320
8.211
-2.480
.186
a. Dependent Variable: Mean total SAT in 1994-1995
58
Standardized Coefficients t
Beta
-.887
Sig.
128.278
.000
-13.317
.000
There was an inverse relation between the percent of students taking the SAT and the mean SAT score, r = -.89, F (1, 48) = 177.33, p < .001. If the percent of students taking the SAT is related to the other predictors, it could considerably change their relation with the mean SAT.
Exercise 10.5 Using the Bivariate Correlations procedure, the following table of r values was obtained: No p < .05 Correlations Head Circumference Full-Scale IQ (cm) Full-Scale IQ
Pearson Correlation
1
Sig. (2-tailed) N Head Circumference (cm) Pearson Correlation Sig. (2-tailed)
Pearson Correlation Sig. (2-tailed)
Sig. (2-tailed)
Pearson Correlation Sig. (2-tailed)
.138
-.291
-.063
-.003
.562
.213
.791
.991
20
20
20
20
1
.337
.508
*
.240
.147
.022
.308
.562 20
20
20
-.291
.337
1
.213
.147
20
20
-.063
.508
.791
.022
N Body Weight (kg)
Body Weight (kg)
20
N Total Brain Volume (cm3) Pearson Correlation
Total Brain Volume (cm3)
.138
N Total Surface Area (cm2)
Total Surface Area (cm2)
*
20
20
**
.064
.005
.788
.601
20
20
20
**
1
.208
.601
.005
.379
20
20
20
20
20
-.003
.240
.064
.208
1
.991
.308
.788
.379
20
20
20
20
N *. Correlation is significant at the 0.05 level (2-tailed). **. Correlation is significant at the 0.01 level (2-tailed).
From column 1 or row 1 it can be seen that none of the variables is significantly related to IQ.
59
20
Chapter 11 Exercise 11.1 This statement in the Select Cases procedure omits cases 69, 70, and 94.
($CASENUM NE 69) AND ($CASENUM NE 70) AND ($CASENUM NE 94)
Use the Linear Regression procedure for the analysis.
60
The output requested in the Plots window are shown below:
The plots allow the first question to be answered:
61
1.
Yes, the distributional assumptions appear to have been met. The histogram showing the standardized residuals of HiwayMPG appears to be approximately normal, and the points in the normal P-P plot fall close to the diagonal. The scatterplot of predicted and observed standardized values of HiwayMPG show no serious violations of linearity or homogeneity of variance. However, there appear to be one or two outliers.
Questions 2 and 3 can be answered from the Correlations table: Correlations HiwayMPG Pearson Correlation HiwayMPG
Sig. (1-tailed)
N
Engine
Cylinders
HP
Weight
WheelBase
Length
1.000
-.730
-.643
-.662
-.822
-.470
-.375
Engine
-.730
1.000
.899
.768
.810
.628
.623
Cylinders
-.643
.899
1.000
.747
.705
.525
.531
HP
-.662
.768
.747
1.000
.623
.391
.370
Weight
-.822
.810
.705
.623
1.000
.749
.648
WheelBase
-.470
.628
.525
.391
.749
1.000
.865
Length
-.375
.623
.531
.370
.648
.865
1.000
.
.000
.000
.000
.000
.000
.000
HiwayMPG Engine
.000
.
.000
.000
.000
.000
.000
Cylinders
.000
.000
.
.000
.000
.000
.000
HP
.000
.000
.000
.
.000
.000
.000
Weight
.000
.000
.000
.000
.
.000
.000
WheelBase
.000
.000
.000
.000
.000
.
.000
Length
.000
.000
.000
.000
.000
.000
.
HiwayMPG
386
386
386
386
386
386
386
Engine
386
386
386
386
386
386
386
Cylinders
386
386
386
386
386
386
386
HP
386
386
386
386
386
386
386
Weight
386
386
386
386
386
386
386
WheelBase
386
386
386
386
386
386
386
Length
386
386
386
386
386
386
386
2. r values for HiwayMPG and each predictor variable -.730 Engine -.643 Cylinders -.662 HP -.822 Weight -.470 WheelBase -.375 Length
62
3.
The two predictors that are most strongly related are Cylinders and Engine (r = .899).
Output needed to answer question 4 is shown below: Model Summaryb Model
R .872a
1
Adjusted R Square
R Square .760
Std. Error of the Estimate
.756
2.47007
a. Predictors: (Constant), Length, HP, Weight, Cylinders, WheelBase, Engine b. Dependent Variable: HiwayMPG
ANOVAb Model 1
Sum of Squares
df
Mean Square
F
Regression
7315.374
6
1219.229
Residual
2312.367
379
6.101
Total
9627.741
385
Sig. .000a
199.833
a. Predictors: (Constant), Length, HP, Weight, Cylinders, WheelBase, Engine b. Dependent Variable: HiwayMPG
4.
For all six predictors combined, R2 = .760, a value that is highly significant, F (6, 379) = 199.833, p < .001.
The Coefficients table gives information for questions 5 and 6: Coefficientsa Unstandardized Coefficients Model 1
B (Constant)
Standardized Coefficients
Std. Error
28.154
2.431
-.970
.368
.265
HP Weight
Correlations t
Beta
Sig.
Zero-order
Partial
11.580
.000
-.197
-2.639
.009
-.730
-.134
-.066
.189
.083
1.401
.162
-.643
.072
.035
-.012
.003
-.173
-4.192
.000
-.662
-.210
-.106
-.006
.000
-.845
-16.361
.000
-.822
-.643
-.412
WheelBase
.105
.042
.148
2.522
.012
-.470
.128
.063
Length
.071
.020
.188
3.579
.000
-.375
.181
.090
Engine Cylinders
a. Dependent Variable: HiwayMPG
5.
Part
HiwayMPG = 28.154 -.970(Engine) +.265(Cylinders) -.012(HP) -.006(Weight) +.105(WheelBase) +.071(Length)
63
6.
The three variables that have the highest semipartial correlations with the dependent variable are Weight, HP, and Length.
Question 7 is answered from information shown in the table below: Casewise Diagnosticsa Case Number 13 47 49 97
341
Make
Std. Residual
Honda Civic HX 2dr Toyota Echo 2dr manual Toyota Echo 4dr Volkswagen Jetta GLS TDI 4dr Chevrolet Tracker
HiwayMPG
Predicted Value
Residual
3.936
44.00
34.2776
9.72235
3.042
43.00
35.4870
7.51302
3.090 6.223
43.00 46.00
35.3663 30.6282
7.63367 15.37181
-3.178
22.00
29.8490
-7.84896
a. Dependent Variable: HiwayMPG
7.
Two cars could be considered outliers (their |std.residual| ≥ 3.3). These are cars 13 and 97, the Honda Civic HX 2dr and the Volkswagen Jetta GLS TDI 4dr.
Exercise 11.3 1.
2.
3.
The prediction equation is Estimated enrollment = -9153.254 + .406 Hgrad + 4.275 Income + 450.125 Unemp. R2 for the equation is .962. To summarize the errors in predicting enrollments using the formula, one could use the standard error of the estimate (which has the value 670.44). Alternatively, and perhaps more appropriate for the question being addressed here, one could use the standard error of the residuals that appears in the Residuals Statistics table (which has the value 633.51). The value is smaller than that of the standard error of the estimate because the denominator of its equation is N – 2 rather than N – k – 1 (in the case of the standard error of the estimate). The value 633.51 is a better answer to our question because the question does not call for generalizing from our sample data to a population value, which is the purpose of the standard error of the estimate. An even better answer to the question could be obtained by saving the unstandardized residuals (available by pressing the Save button on the Linear Regression Window) then transforming these saved scores to their absolute values with a Compute transformation and then reporting a summary measure of their central tendency available in the Explore procedure. The result is a mean absolute deviation of 487.91 (the median absolute deviation is 454.99). Thus, the mean number of students wrongly predicted using this model each year between 1961 and 1989 would be about 488. The administrator’s suggestion could be examined using hierarchical regression in which the dependent variable, enrollment, is assessed in model 1 from the predictor
64
Year (it doesn’t matter if year is coded from 1 to 29 or if it is represented as 1961 to 1989). In model 2, the effect on R2 of introducing the other three variables is assessed. The tables below show the results of these steps: Model Summaryc Change Statistics Model 1 2
R R Square .901a .812 .986b .971
Adjusted R Square .805 .967
Std. Error of the Estimate 1438.03744 594.29952
R Square Change .812 .160
F Change 116.375 44.695
df1
df2 1 3
27 24
Sig. F Change .000 .000
a. Predictors: (Constant), Year data obtained from 1961-1989 recoded as 1-29 b. Predictors: (Constant), Year data obtained from 1961-1989 recoded as 1-29, January unemployment rate (%) for New Mexico , Spring high schoolgraduates in New Mexico , Per capita income in Albuquerque (1961 dollars) c. Dependent Variable: Fall undergraduate enrollments, U of New Mexico
ANOVAc Model 1
2
Regression Residual Total Regression Residual Total
Sum of Squares 2.4E+008 55834696 3.0E+008 2.9E+008 8476606 3.0E+008
df 1 27 28 4 24 28
Mean Square 240657781.4 2067951.688
F 116.375
Sig. .000a
72003967.73 353191.918
203.866
.000b
a. Predictors: (Constant), Year data obtained from 1961-1989 recoded as 1-29 b. Predictors: (Constant), Year data obtained from 1961-1989 recoded as 1-29, January unemployment rate (%) for New Mexico , Spring high schoolgraduates in New Mexico , Per capita income in Albuquerque (1961 dollars) c. Dependent Variable: Fall undergraduate enrollments, U of New Mexico
For model 2, R2 change is .16, a value that differs significantly from chance, F(3, 24) = 44.70, p < .001. Thus, including the variables Unemp, Hgrad, and Income significantly improves predicted enrollment over and above the linear effect of Year.
Exercise 11.5 1.
The parameters of the equation for the males are shown in the table below: Unstandardized Regression Coefficient -115.948 1.809 1.805 .961 1.228 .198 .906
Variable (Constant) Chest diameter Chest depth Bitrochanteric diameter Ankle diameter Height Knee diameter
For these data, R2 = .82, F(6, 240) = 177.11, p < .001.
65
2.
The parameters of the equation for the females are shown in the table below: Unstandardized Regression Coefficient -111.257 2.899 1.551 1.156 .199 .706 -1.731 1.947 .366
Variable (Constant) Knee diameter Chest depth Chest diameter Height Bitrochanteric diameter Ankle diameter Wrist diameter Biiliac diameter, (pelvic breadth)
For these data, R2 = .81, F(8, 251) = 133.11, p < .001. 3.
For males and females, the 4-term equations are:
Variable (Constant) Chest girth Hip girth Height
Males Unstandardized Regression Coefficient -131.633 .527 .914 .378
Females Unstandardized Regression Coefficient -110.809 .602 .774 .276
For males, the standard error is 3.34. For females the standard error is 2.91. Note: Convert inches to cm by multiplying by 2.54; convert lbs to kg by multiplying by .45.
66
Chapter 12 Exercise 12.1 The question may be answered with a chi-square test of independence. The data can be represented by identifying 4 cells in a 2 x 2 table and weighting each by a frequency count as shown below:
Tables that show important output of the analysis are displayed below: Still dating? * Writing condition Crosstabulation Writing condition Emotional Still dating?
Yes
Count
Total
Total
34
22
56
28.7
27.3
56.0
Residual
5.3
-5.3
Count
10
20
30
Expected Count
15.3
14.7
30.0
Residual
-5.3
5.3
44
42
86
44.0
42.0
86.0
Expected Count
No
Control
Count Expected Count
Chi-Square Tests Value
Asymp. Sig. (2sided)
df
5.861a
1
.015
Continuity Correction
4.817
1
.028
Likelihood Ratio
5.943
1
.015
Pearson Chi-Square b
Exact Sig. (2sided)
Fisher's Exact Test Linear-by-Linear Association N of Valid Cases
.023 5.793
1
.016
86
a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 14.65. b. Computed only for a 2x2 table
67
Exact Sig. (1sided)
.014
Symmetric Measures Value Nominal by Nominal
Approx. Sig.
Phi
.261
.015
Cramer's V
.261
.015
N of Valid Cases
86
The analysis shows that emotional writing is significantly related to relationship stability, χ2 (1, N = 86) = 5.86, p < .025, φ = .26. The value of phi indicates a medium effect size for the relationship.
Exercise 12.3 The problem calls for a chi-square goodness-of-fit test. The theoretical values are the percents given for the U.S. in each speed category. Data entry is shown below:
The value labels for SpeedType are shown below:
68
The data are first weighted by the frequency counts of the variable California:
Then a chi-square goodness-of-fit test is performed after entering the values of the U.S. percentages for each category of highway speed:
69
These tables were produced: Speed category of road Observed N .01, w = .14.
Exercise 12.5 The data can be read directly from the file HappyMarried.sav.
These are the variable names that are assigned to rows and columns.
70
These are the tables that are produced: General Happiness * Marital Status Crosstabulation Marital Status Married General Happiness
Very happy
Count Expected Count
Separated
Never married
Total
264
3
32
5
71
375
11.9
58.1
14.7
95.0
375.0
68.7
-8.9
-26.1
-9.7
-24.0
Count
316
24
119
25
188
672
Expected Count
350.0
21.3
104.2
26.3
170.2
672.0
Residual
-34.0
2.7
14.8
-1.3
17.8
45
11
35
17
45
153 153.0
Not too happy Count Expected Count Residual Total
Divorced
195.3
Residual Pretty happy
Widowed
79.7
4.8
23.7
6.0
38.8
-34.7
6.2
11.3
11.0
6.2
625
38
186
47
304
1200
625.0
38.0
186.0
47.0
304.0
1200.0
Count Expected Count
Chi-Square Tests Value Pearson Chi-Square Likelihood Ratio Linear-by-Linear Association N of Valid Cases
112.180a 110.902 52.891 1200
Asymp. Sig. (2sided)
df 8 8 1
.000 .000 .000
a. 1 cells (6.7%) have expected count less than 5. The minimum expected count is 4.85.
Symmetric Measures Value Nominal by Nominal
Approx. Sig.
Phi
.306
.000
Cramer's V
.216
.000
N of Valid Cases
1200
The analysis shows that marital status is significantly related to general happiness, χ2 (8, N = 1,200) = 112.18, p < .001, w = .31. The value of w indicates a medium effect size for the relationship.
71
Chapter 13 Exercise 13.1 1.
Here is the Recode window that can produce the needed transformations:
2.
The Explore procedure indicates that the distribution of the variable mem1 may not be normal for the low and high suggestibility group due to the presence of too many low scores. Histograms and Q-Q plots are shown below for each of these groups:
72
3.
The new variable mem2 is created using the Compute Variable procedure using the formula mem2 = mem1 * mem1. The Explore procedure for the new variable mem2 shows the assumption of normality is better met.
4.
The results of the t-test for independent samples is shown below:
73
Group Statistics group mem2
N
Mean
Std. Deviation
Std. Error Mean
1.00
32
.3422
.23731
.04195
2.00
22
.4364
.23165
.04939
Independent Samples Test Levene's Test for Equality of Variances
F mem2
5.
Equal variances assumed Equal variances not assumed
.000
t-test for Equality of Means
Sig.
t
.990
df
Sig. (2-tailed)
Mean Difference
Std. Error Difference
95% Confidence Interval of the Difference Lower Upper
-1.447
52
.154
-.09418
.06510
-.22480
.03645
-1.453
46.009
.153
-.09418
.06480
-.22461
.03626
The outcome of the t-test can be summarized as follows: The corrected recognition scores of the low suggestibility participants (those having objective scores on the Barber Suggestibility Scale of 0.0-2.5) and high suggestibility participants (those with objective scores on the Barber Suggestibility Scale of 5.50-8.00) were transformed to their squared values to better meet the assumption that scores in each sample were normally distributed. A t-test for independent samples was used to determine whether the mean squared corrected recognition score of the low suggestibility participants (M = .34, SD = .24) differed from that of the high suggestibility participants (M = .44, SD = .23). The two means were found to not differ significantly, t (52) = 1.45, p > .10. Thus, it appears that the recognition ability of high and low suggestible participants, as defined and measured in this study, did not differ significantly.
Exercise 13.3 1.
Output of a single sample t-test performed on the variable MPG and using the reference value 55 is shown below: One-Sample Statistics N MPG
74
Mean 47.8743
Std. Deviation 5.14521
Std. Error Mean .59812
One-Sample Test Test Value = 55
MPG
t -11.913
df 73
Sig. (2-tailed) .000
74
Mean Difference -7.12568
95% Confidence Interval of the Difference Lower Upper -8.3177 -5.9336
Using Explore, a histogram and normal Q-Q plot was obtained to check the assumption that the variable MPG was approximately normally distributed: Histogram
Normal Q-Q Plot of MPG
3 12.5
2
Expected Normal
Frequency
10.0
7.5
5.0
1
0
-1
-2
2.5 Mean =47.87 Std. Dev. =5.145 N =74
0.0 35.00
40.00
45.00
50.00
55.00
-3
60.00
35
MPG
40
45
50
55
60
65
Observed Value
Here is how the outcome of the test can be described: The mean combined city/highway mileage of 74 2006 Prius owners was found to be 47.87 (SD = 5.15). From an informal analysis of the variable using a histogram and normal Q-Q plot no serious threats to the assumption of normality were apparent. A single-sample t-test revealed that the mean mpg differed significantly from 55, the combined EPA city/highway mpg estimate, t (73) = 11.91, p = .001. The effect size as measured by d was 1.39, a value that can be considered large. Thus, the actual combined city/highway mileage of the 2006 Prius is significantly different the EPA estimate of 55mpg. 2.
The actual percent of highway driving differs significantly from the value 45 used by the EPA. Output of the single sample t-test is shown below: One-Sample Statistics N Hiway
69
Mean 58.30
Std. Deviation 23.963
Std. Error Mean 2.885
One-Sample Test Test Value = 45
Hiway
t 4.612
df 68
Sig. (2-tailed) .000
75
Mean Difference 13.304
95% Confidence Interval of the Difference Lower Upper 7.55 19.06
Exercise 13.5 1.
The variable Dfry was created using the following Compute statement:
2.
Data for Pennsylvania (state = 0) were selected using the Select Cases: If window:
A single sample t-test performed on the variable Dfry using the reference value 0:
76
The following output was produced: One-Sample Statistics N Dfry
71
Mean .0192
Std. Deviation .06212
Std. Error Mean .00737
One-Sample Test Test Value = 0
Dfry
t 2.598
df 70
Sig. (2-tailed) .011
95% Confidence Interval of the Difference Lower Upper .0045 .0339
Mean Difference .01915
There was a significant increase in the price of fries. 3.
A similar analysis was performed for the restaurants in New Jersey: One-Sample Statistics N Dfry
295
Mean .0168
Std. Deviation .07004
Std. Error Mean .00408
One-Sample Test Test Value = 0
Dfry
t 4.131
df 294
Sig. (2-tailed) .000
Mean Difference .01685
Again, there was a significant increase in the price of fries.
77
95% Confidence Interval of the Difference Lower Upper .0088 .0249
4.
A t-test for independent samples was performed for the restaurants in Pennsylvania and New Jersey to compare the value of the variable Dfry.
Group Statistics
Dfry
State Pennsylvania New Jersey
N
Mean .0192 .0168
71 295
Std. Deviation .06212 .07004
Std. Error Mean .00737 .00408
Independent Samples Test Levene's Test for Equality of Variances
Dfry
Equal variances assumed Equal variances not assumed
t-test for Equality of Means
F
Sig.
t
1.147
.285
.254 .274
95% Confidence Interval of the Difference Lower Upper
Sig. (2-tailed)
Mean Difference
Std. Error Difference
364
.799
.00231
.00907
-.01552
.02014
116.784
.785
.00231
.00843
-.01438
.01899
df
The mean value of Dfry did not differ for the two states, t < 1. This may indicate that the increase in the price of a small fry that appeared related to an increase in minimum wage in New Jersey was due to some other, perhaps more general, cause (and not the increase in minimum wage).
78
Chapter 14 Exercise 14.1 1.
The data were labeled and values entered for the variable Major:
2.
Cases were selected using the IF option from the Select Cases window:
79
3.
A one-way ANOVA was run using the following windows:
80
Here is the output: Descriptives Expected grade in class 95% Confidence Interval for Mean N criminal justice exercise science nursing psychology Total
Mean 17 11 25 12 65
Std. Deviation Std. Error Lower Bound Upper Bound Minimum Maximum
3.029 3.709 3.608 3.500 3.454
.5818 .3562 .4051 .2828 .4985
.1411 .1074 .0810 .0816 .0618
2.730 3.470 3.441 3.320 3.330
3.329 3.948 3.775 3.680 3.577
2.0 3.0 2.5 3.0 2.0
Test of Homogeneity of Variances Expected grade in class Levene Statistic
df1
df2
1.953
3
Sig. 61
.131
ANOVA Expected grade in class Sum of Squares
df
Mean Square
Between Groups Within Groups
4.399 11.503
3 61
Total
15.902
64
81
1.466 .189
F 7.776
Sig. .000
4.0 4.0 4.0 3.9 4.0
4.
The results of the analysis are summarized below: Differences among the mean expected grades of students in four majors in college (see Table 14.2 below) were assessed with a one-way ANOVA. A Levene test of homogeneity of variance conducted prior to the ANOVA did not indicate the assumption of homogeneity of variance was significantly violated (p > .10). The ANOVA revealed the differences among the mean expected grade of the four majors was significant, F (3, 61) = 7.77, p < .001, η2 = .28. Table 14.2. Descriptive statistics for ANOVA example. Standard Major N Mean Deviation Criminal Justice 17 3.03 0.58 Exercise Science 11 3.71 0.36 Nursing 25 3.61 0.41 Psychology 12 3.50 0.28
82
5.
This question can be addressed with a Dunnett test (or a Tukey test). A Dunnett test is appropriate because the same group is being contrasted to every other individual group:
The table shown below gives the outcome of the test: Multiple Comparisons Expected grade in class Dunnett t (2-sided)a (I) Major in college
(J) Major in college
Mean Difference (I-J) Std. Error
95% Confidence Interval Sig.
Lower Bound Upper Bound
.6797
*
.1680
.000
.274
1.086
.5786
*
.1365
.000
.249
.908
psychology criminal justice .4706 .1637 .015 .075 a. Dunnett t-tests treat one group as a control, and compare all other groups against it.
.866
exercise science criminal justice nursing
criminal justice
*
*. The mean difference is significant at the 0.05 level.
The analysis shows that the mean expected grade of the criminal justice majors differs significantly from that of every other individual major in the study (all ps < .025). 6.
The outcome of the one-way ANOVA performed on the dependent variable Number is shown below:
83
Descriptives Number of correct answers
criminal justice exercise science nursing psychology Total
N 18 11 27 12 68
Mean 8.0000 8.1818 7.9259 8.0000 8.0000
Std. Deviation 2.14202 1.32802 2.60068 1.53741 2.10897
Std. Error .50488 .40041 .50050 .44381 .25575
95% Confidence Interval for Mean Lower Bound Upper Bound 6.9348 9.0652 7.2896 9.0740 6.8971 8.9547 7.0232 8.9768 7.4895 8.5105
Minimum 3.00 6.00 2.00 6.00 2.00
Test of Homogeneity of Variances Number of correct answers Levene Statistic 2.955
df1
df2 3
64
Sig. .039
ANOVA Number of correct answers
Between Groups Within Groups Total
Sum of Squares .512 297.488 298.000
df 3 64 67
84
Mean Square .171 4.648
F .037
Sig. .991
Maximum 11.00 10.00 12.00 10.00 12.00
The results show that the mean number of correct answers on the statistical concepts test did not differ significantly among the four majors, F < 1.
Exercise 14.3 1.
The data are entered in two columns as shown below (for a subset of the cases):
Suitable variable names and value labels are created:
The analysis is requested with the following windows:
85
This output is produced: Descriptives Nwords 95% Confidence Interval for Mean N High School Young Adult Middle Aged Retired Total
Mean 7 7 7 7 28
Std. Deviation Std. Error Lower Bound
3.8571 3.7143 6.2857 7.4286 5.3214
1.57359 1.38013 1.11270 1.71825 2.12661
.59476 .52164 .42056 .64944 .40189
Upper Bound
2.4018 2.4379 5.2566 5.8395 4.4968
5.3125 4.9907 7.3148 9.0177 6.1460
Test of Homogeneity of Variances Nwords Levene Statistic .493
df1
df2 3
ANOVA Nwords
86
Sig. 24
.690
Minimum Maximum 2.00 2.00 5.00 5.00 2.00
6.00 6.00 8.00 10.00 10.00
Sum of Squares Between Groups Within Groups Total
1. 2.
df
Mean Square
70.679 51.429
3 24
122.107
27
23.560 2.143
F 10.994
Sig. .000
The Levene test of homogeneity of variance is not significant, so the assumption of homogeneity of variance is not rejected. The highly significant F value indicates that the mean number of words spelled correctly is significantly affected by age. Testing if high school students correctly spell a significantly different number of words correctly than every other individual age group calls for a Dunnett test (a Tukey HSD test would also be acceptable). The window that requests this test from the Post Hoc tests is shown below:
87
The data produced by this test is shown next: Multiple Comparisons Nwords Dunnett t (2-sided)a (J) Age Category
Young Adult
High School
-.14286
.78246
.996
-2.1042
1.8185
Middle Aged
High School
2.42857*
.78246
.013
.4672
4.3899
High School
*
.78246
.000
1.6101
5.5328
Retired
Mean Difference (I-J)
95% Confidence Interval
(I) Age Category
Std. Error
3.57143
Sig.
Lower Bound
Upper Bound
a. Dunnett t-tests treat one group as a control, and compare all other groups against it. *. The mean difference is significant at the 0.05 level.
The test reveals that the mean number of words spelled correctly by the high school students differs significantly from that of the middle aged (p < .025) and the retired groups (p < .001).
88
Chapter 15 Exercise 15.1 1.
The data were coded into SPSS by creating three variables:
2.
The 2-way ANOVA was run from the General Linear Model using the windows shown below:
89
The output is shown below: Between-Subjects Factors Value Label
N
Factor A
1.00
a1
10
Factor B
2.00 1.00
a2 b1
10 10
2.00
b2
10
90
Descriptive Statistics Dependent Variable:Dependent variable Factor A Factor B a1
a2
Mean
N
b1
6.2000
1.30384
5
b2
16.0000
2.34521
5
Total
11.1000
5.46606
10
b1
11.8000
1.30384
5
b2
9.0000
1.58114
5
10.4000
2.01108
10
b1
9.0000
3.19722
10
b2
12.5000
4.14327
10
Total
10.7500
4.02460
20
Total Total
Std. Deviation
a
Levene's Test of Equality of Error Variances Dependent Variable:Dependent variable F
df1 2.228
df2 3
Sig. 16
.124
Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a. Design: Intercept + A + B + A * B
Tests of Between-Subjects Effects Dependent Variable:Dependent variable Source
Type III Sum of Squares
df
Mean Square
F
Sig.
Partial Eta Squared
Corrected Model
262.150
a
3
87.383
30.661
.000
.852
Intercept
2311.250
1
2311.250
810.965
.000
.981
A
2.450
1
2.450
.860
.368
.051
B
61.250
1
61.250
21.491
.000
.573
A*B
198.450
1
198.450
69.632
.000
.813
Error
45.600
16
2.850
Total
2619.000
20
307.750
19
Corrected Total
a. R Squared = .852 (Adjusted R Squared = .824)
91
The plot was edited to look like this:
The results can be described like this: The effect of factor A (a1, a2) and factor B (b1, b2) on mean values of the dependent variable was examined using a two-way analysis of variance (ANOVA) for independent groups. Descriptive statistics for the dependent variable as a function of levels of the two factors are shown in Table 15.4. The ANOVA revealed the interaction of factors A and B was significant, F (1, 16) = 69.63, p < .001, partial η2 = .81. In addition, there was a significant main effect of factor B, F (1, 16) = 21.49, p < .001, partial η2 = .57. The main effect of factor A was not significant, F < 1.
92
Table 15.4. Descriptive statistics for the dependent variable number of words recalled as a function of cue condition and cognitive load. Mean Dependent Variable Values as a Function of Factors A and B Factor A Factor B Mean Std. Deviation N a1 b1 6.2000 1.30384 5 b2 16.0000 2.34521 5 Total 11.1000 5.46606 10 a2 b1 11.8000 1.30384 5 b2 9.0000 1.58114 5 Total 10.4000 2.01108 10 Total b1 9.0000 3.19722 10 b2 12.5000 4.14327 10 Total 10.7500 4.02460 20
Exercise 15.3 The arrows show the means compared for each contrast type:
Deviation
b1 b2
last
b1 b2 1 2 3 4
Simple
first
b1 b2
Grand mean
1 2 3 4 b1 b2
1 2 3 4
Difference
none
none
b1 b2
Repeated
none
b1 b2
b1 b2 n
1 2 3 4
b1 b2 1 2 3 4
b1 b2 1 2 3 4
1 2 3 4
1 2 3 4
b1 b2
b1 b2
1 2 3 4 b1 b2
b1 b2
1 2 3 4
Grand mean
1 2 3 4
1 2 3 4
Helmert
b1 b2
1 2 3 4 b1 b2
1 2 3 4
93
1 2 3 4
Grand mean
Chapter 16 Exercise 16.1 A one-way within-subjects ANOVA was performed to answer the questions in parts 1 and 2 using the variables Oct2004, Oct2005, and Oct2006 as the three variables. The following selections were made to implement the analysis:
Any variable name can be specified here.
Press Add then Define.
94
Transfer these variables by pressing this button.
95
Transfer this factor to the Horizontal Axis then press Add.
The output produced is shown below: Within-Subjects Factors Measure:MEASURE_1 Dependent Variable
time 1
Oct2004
2
Oct2005
3
Oct2006
Descriptive Statistics Mean
Std. Deviation
N
Number of reports of mishandled baggage per 1000 passengers in 2004
4.4794
2.78175
17
Number of reports of mishandled baggage per 1000 passengers in 2005
5.3800
2.69124
17
Number of reports of mishandled baggage per 1000 passengers in 2006
8.4818
4.99308
17
b
Multivariate Tests Effect time
Value
F
Hypothesis df 2.000
15.000
.000
.761
a
2.000
15.000
.000
.761
a
2.000
15.000
.000
.761
a
2.000
15.000
.000
.761
.761
23.837
Wilks' Lambda
.239
23.837
Hotelling's Trace
3.178
23.837
Roy's Largest Root
3.178
23.837
b. Design: Intercept Within Subjects Design: time
96
Partial Eta Squared
Sig.
a
Pillai's Trace
a. Exact statistic
Error df
Mauchly's Test of Sphericityb Measure:MEASURE_1 Within Subjec ts Effect Mauchly's W
Epsilona Approx. ChiSquare
df
Sig.
GreenhouseGeisser
Huynh-Feldt Lower-bound
time .452 11.898 2 .003 .646 .679 .500 Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. b. Design: Intercept Within Subjects Design: time Tests of Within-Subjects Effects Measure:MEASURE_1 Type III Sum of Squares
Source time
Error(time)
df
Mean Square
F
Sig.
Partial Eta Squared
Sphericity Assumed
149.888
2
74.944
32.309
.000
.669
Greenhouse-Geisser
149.888
1.292
115.985
32.309
.000
.669
Huynh-Feldt
149.888
1.358
110.395
32.309
.000
.669
Lower-bound
149.888
1.000
149.888
32.309
.000
.669
Sphericity Assumed
74.227
32
2.320
Greenhouse-Geisser
74.227
20.677
3.590
Huynh-Feldt
74.227
21.724
3.417
Lower-bound
74.227
16.000
4.639
Tests of Within-Subjects Contrasts Measure:MEASURE_1 Source
time
time
Linear Quadratic
Error(time) Linear Quadratic
Type III Sum of Squares
df
Mean Square
F
Sig.
Partial Eta Squared
136.160
1
136.160
44.615
.000
.736
13.728
1
13.728
8.649
.010
.351
48.831
16
3.052
25.396
16
1.587
97
Tests of Between-Subjects Effects Measure:MEASURE_1 Transformed Variable:Average Source
Type III Sum of Squares
Intercept Error
1.
2.
1906.260 564.362
df
Mean Square 1 16
1906.260 35.273
F 54.044
Sig. .000
Partial Eta Squared .772
Because Mauchly’s test of sphericity is significant, the effect of time could be assessed using either a univariate F based on corrected degrees of freedom (the Huynh-Feldt,{ XE "Huynh-Feldt correction" } Greenhouse-Geisser{ XE "Greenhouse-Geisser correction" }, or lower-bound procedure) or a multivariate statistic. Both approaches indicate the effect of time was highly significant. The linear effect of time is highly significant, F (1, 16) = 44.62, p < .001, partial η2 = .74.
Exercise 16.3 The data should be analyzed as a mixed two-way ANOVA. The between-subjects factor is given the name condition (new, standard) and the within-subjects factor is called display (time1, time2, time3). The question calls for analysis of the main effect of the
98
between-subjects factor, condition. The selections used to analyze the data are shown below:
Press Add then Define.
Transfer these highlighted variables by pressing this button. Use condition as the betweensubjects factor.
99
Output produced for the analysis is shown next: Within-Subjects Factors Measure:MEASURE_1 display 1 2 3
Dependent Variable time1 time2 time3
100
Between-Subjects Factors Value Label processing condition
N
1
new
60
2
control
60
Descriptive Statistics processing condition time1
time2
time3
Mean
Std. Deviation
N
new
232.15
82.854
60
control
247.53
100.981
60
Total
239.84
92.298
120
new
238.02
123.236
60
control
436.68
147.955
60
Total
337.35
168.325
120
new
401.78
128.641
60
control
299.27
131.827
60
Total
350.53
139.536
120
b
Multivariate Tests Effect
Value
display
display * condition
F
Hypothesis df
Error df
Sig.
Partial Eta Squared
a
2.000
117.000
.000
.514
a
2.000
117.000
.000
.514
a
2.000
117.000
.000
.514
a
2.000
117.000
.000
.514
a
2.000
117.000
.000
.585
a
2.000
117.000
.000
.585
a
2.000
117.000
.000
.585
a
2.000
117.000
.000
.585
Pillai's Trace
.514
61.875
Wilks' Lambda
.486
61.875
Hotelling's Trace
1.058
61.875
Roy's Largest Root
1.058
61.875
Pillai's Trace
.585
82.530
Wilks' Lambda
.415
82.530
Hotelling's Trace
1.411
82.530
Roy's Largest Root
1.411
82.530
a. Exact statistic b. Design: Intercept + condition Within Subjects Design: display
b
Mauchly's Test of Sphericity Measure:MEASURE_1 Within Subjects Effect display
a
Epsilon Mauchly's W .984
Approx. ChiSquare 1.889
df
GreenhouseGeisser
Sig. 2
.389
.984
Huynh-Feldt 1.000
Lower-bound .500
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. b. Design: Intercept + condition Within Subjects Design: display
101
Tests of Within-Subjects Effects Measure:MEASURE_1 Type III Sum of Squares
Source display
Error(display)
Mean Square
F
Partial Eta Squared
Sig.
Sphericity Assumed
877290.239
2
438645.119
56.781
.000
.325
GreenhouseGeisser
877290.239
1.968
445669.175
56.781
.000
.325
Huynh-Feldt
877290.239
2.000
438645.119
56.781
.000
.325
Lower-bound display * condition
df
877290.239
1.000
877290.239
56.781
.000
.325
Sphericity Assumed
1382045.906
2
691022.953
89.451
.000
.431
GreenhouseGeisser
1382045.906
1.968
702088.353
89.451
.000
.431
Huynh-Feldt
1382045.906
2.000
691022.953
89.451
.000
.431
Lower-bound
1382045.906
1.000 1382045.906
89.451
.000
.431
Sphericity Assumed
1823135.856
236
7725.152
GreenhouseGeisser
1823135.856
232.280
7848.855
Huynh-Feldt
1823135.856
236.000
7725.152
Lower-bound
1823135.856
118.000
15450.304
Tests of Within-Subjects Contrasts Measure:MEASURE_1 Type III Sum of Squares
display
display
Linear
735048.017
1
735048.017
108.920
.000
.480
Quadratic
142242.222
1
142242.222
16.346
.000
.122
display * condition
Linear
208506.150
1
208506.150
30.897
.000
.208
1 1173539.756 134.862
.000
.533
Error(display)
Linear
Quadratic Quadratic
df
1173539.756
Mean Square
796321.833
118
6748.490
1026814.022
118
8701.814
F
Sig.
Partial Eta Squared
Source
Tests of Between-Subjects Effects Measure:MEASURE_1 Transformed Variable:Average Source Intercept condition Error
Type III Sum of Squares 3.443E7 124396.844 3372790.611
df
Mean Square 1 1 118
3.443E7 124396.844 28582.971
102
F 1204.435 4.352
Sig. .000 .039
Partial Eta Squared .911 .036
Estimated Marginal Means processing condition Measure:MEASURE_1 processin g condition new control
95% Confidence Interval Mean 290.650 327.828
Std. Error 12.601 12.601
Lower Bound 265.696 302.874
Upper Bound 315.604 352.782
From the table labeled Tests of Between-subjects Effects the effect of condition is found to be significant, F (1, 118) = 4.35, p < .05, partial η2 = .04.
103
Chapter 17 Exercise 17.1 To test this question, a binomial test could be performed. Here, the number of hurricanes is 7 and the number of non-hurricanes is 20. These data could be entered in the SPSS Data Editor in the following way:
Then, the values of the numeric variable StormType would be weighted by values of the variable Number.
The binomial test would be run using the test proportion is .248.
104
The following table indicates the proportion of hurricanes vs. non-hurricanes during 2005 did not differ from the historical proportion. Descriptive Statistics N StormType
Mean 27
Std. Deviation
1.7407
Minimum
.44658
Maximum
1.00
2.00
Binomial Test Category StormType
Group 1
hurricane
Group 2
not hurricane
N
Total
Observed Prop. 7
.259
20
.741
27
1.000
Test Prop. .248
Asymp. Sig. (1-tailed) .520a
a. Based on Z Approximation.
A description of the outcome is shown below: Historically, since records were begun in 1944, the proportion of Atlantic storms classified as hurricanes during the hurricane season has been .248. A binomial test was used to determine whether the number of storms classified as hurricanes during the 2005 hurricane season differed significantly from this value (Nhurricane = 7, Nnot hurricane = 20). The test revealed that the proportion of hurricanes during 2005 did not differ significantly from the historical proportion of .248, p > .25.
Exercise 17.3 The question may be addressed with a Mann-Whitney U test. The data can be read from the file HappyMarried.sav. The two groups are defined by selecting the values 1 and 5 for the variable marital.
105
These tables are produced: Ranks Marital Status General Happiness
N
Mean Rank
Sum of Ranks
Married
625
431.69
269804.50
Never married
304
533.49
162180.50
Total
929
Test Statisticsa General Happiness Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed)
74179.500 269804.500 -6.095 .000
a. Grouping Variable: Marital Status
The results could be summarized as follows: Using a Mann-Whitney U test, the two distributions of happiness ratings (mean rankmarried = 431.69, mean ranknever married = 533.49) were found to differ significantly, U = 74179.5, z = 6.10, p < .001, A = .61. The A value .61 corresponds to a small to medium effect (Vargha and Delaney, 2000). Note that a lower mean happiness rank corresponds to greater judged happiness.
Exercise 17.5 The question can be addressed with a Mann-Whitney U test.
106
The following tables are produced: Ranks processin g condition time
N
Mean Rank
Sum of Ranks
new
22
14.64
322.00
control
22
30.36
668.00
Total
44
Test Statisticsa time Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed)
69.000 322.000 -4.061 .000
a. Grouping Variable: processing condition
The table labeled Ranks indicates that reading times in the control condition are greater than those in the new condition. The z value is large and its probability is low, indicating the distribution of reading times for the new and control instruction conditions differ significantly. The effect size, A is Acontrol ,new
⎛ 668 23 ⎞ − ⎟ ⎜ 22 2⎠ ⎝ = = .86 22
107
The results could be summarized as follows: Using a Mann-Whitney U test, the two distributions of ratings (mean ranknew = 14.64, mean rankcontrol = 30.36) were found to differ significantly, U = 69, z = 4.06, p < .001, A = .86. The A value .86 corresponds to a large effect (Vargha and Delaney, 2000).
Exercise 17.7 Because the data are paired, that is, each participant provides a reading time for the first and the second page, the data may be analyzed using a Wilcoxon signed ranks test. The window that requests the analysis is shown below:
The output produced consists of the following tables: Ranks N time2 - time1
Mean Rank 13.00
65.00
b
32.09
1765.00
Negative Ranks
5
Positive Ranks
55
c
Ties
0
Total
60
a. time2 < time1 b. time2 > time1 c. time2 = time1
b
Test Statistics
time2 - time1 a
Z
-6.257
Asymp. Sig. (2-tailed) a. Based on negative ranks. b. Wilcoxon Signed Ranks Test
108
Sum of Ranks
a
.000
The analysis reveals that there is a significant difference between the median reading times for the two pages. The median reading time for each page mentioned in the summary shown below was obtained using the Explore procedure. The median reading time for page one was 22.45 seconds and that for page two was 42.65 seconds. This difference, tested with a Wilcoxon signed ranks test was significant, z = 6.26, p < .001. For this test, the smaller sum of ranks was 65.00, N = 5, and there were no tied ranks. The A value for the effect was .92, which corresponds to a large effect (Vargha and Delaney, 2000).
109
Chapter 18 Exercise 18.1 These are the screens that were used to analyze the data:
110
Here is the output produced:
111
Descriptive Statistics People living in cities (%) Average female life expectancy Average male life expectancy People who read (%) Population increase (% per year)) Infant mortality (deaths per 1000 live births) Gross domestic product / capita Birth rate per 1000 people Death rate per 1000 people Birth to death ratio Fertility: average number of kids cropgrow Males who read (%) Females who read (%)
Mean 53.18
Std. Deviation 24.379
Analysis N 84
67.71
10.751
84
62.80
9.432
84
73.35
23.308
84
1.999
1.1132
84
51.607
38.3961
84
3730.95
4719.432
84
29.119
11.8499
84
9.60
4.728
84
3.6016
2.11883
84
3.958
1.8857
84
17.61 78.48 66.87
15.930 20.434 28.551
84 84 84
Correlation Matrix
Correlation
People living in cities (%) Average female life expectancy Average male life expectancy People who read (%) Population increase (% per year)) Infant mortality (deaths per 1000 live births) Gross domestic product / capita Birth rate per 1000 people Death rate per 1000 people Birth to death ratio Fertility: average number of kids cropgrow Males who read (%) Females who read (%)
Average male life expectancy .731
People who read (%) .620
Population increase (% per year)) -.254
Infant mortality (deaths per 1000 live births) -.725
Gross domestic product / capita .591
Birth rate per 1000 people -.596
Death rate per 1000 people -.588
Birth to death ratio .174
Fertility: average number of kids -.565
cropgrow -.213
Males who read (%) .590
Females who read (%) .615
People living in cities (%) 1.000
Average female life expectancy .749
.749
1.000
.979
.837
-.470
-.955
.575
-.838
-.793
.121
-.815
.062
.775
.817
.731
.979
1.000
.768
-.370
-.924
.564
-.766
-.839
.211
-.750
.039
.715
.744
.620
.837
.768
1.000
-.624
-.880
.452
-.842
-.550
-.112
-.837
.144
.947
.973
-.254
-.470
-.370
-.624
1.000
.489
-.366
.806
-.007
.743
.792
-.445
-.614
-.633
-.725
-.955
-.924
-.880
.489
1.000
-.586
.838
.727
-.089
.807
-.097
-.808
-.843
.591
.575
.564
.452
-.366
-.586
1.000
-.586
-.263
-.131
-.505
.020
.418
.430
-.596
-.838
-.766
-.842
.806
.838
-.586
1.000
.471
.320
.972
-.262
-.791
-.832
-.588
-.793
-.839
-.550
-.007
.727
-.263
.471
1.000
-.551
.494
.175
-.487
-.512
.174
.121
.211
-.112
.743
-.089
-.131
.320
-.551
1.000
.296
-.395
-.144
-.138
-.565
-.815
-.750
-.837
.792
.807
-.505
.972
.494
.296
1.000
-.219
-.793
-.837
-.213 .590 .615
.062 .775 .817
.039 .715 .744
.144 .947 .973
-.445 -.614 -.633
-.097 -.808 -.843
.020 .418 .430
-.262 -.791 -.832
.175 -.487 -.512
-.395 -.144 -.138
-.219 -.793 -.837
1.000 .178 .173
.178 1.000 .964
.173 .964 1.000
KMO and Bartlett's Test Kaiser-Meyer-Olkin Measure of Sampling Adequacy. Bartlett's Test of Sphericity
Approx. Chi-Square df Sig.
112
.865 2010.834 91 .000
Communalities People living in cities (%) Average female life expectancy Average male life expectancy People who read (%) Population increase (% per year)) Infant mortality (deaths per 1000 live births) Gross domestic product / capita Birth rate per 1000 people Death rate per 1000 people Birth to death ratio Fertility: average number of kids cropgrow Males who read (%) Females who read (%)
Initial 1.000
Extraction .671
1.000
.953
1.000
.915
1.000
.877
1.000
.927
1.000
.932
1.000
.373
1.000
.932
1.000
.839
1.000
.866
1.000
.894
1.000 1.000 1.000
.406 .809 .861
Extraction Method: Principal Component Analysis. Total Variance Explained
Component 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Total 8.658 2.597 .929 .707 .432 .267 .131 .095 .057 .046 .036 .020 .019 .007
Initial Eigenvalues % of Variance Cumulative % 61.841 61.841 18.550 80.391 6.637 87.027 5.049 92.077 3.082 95.159 1.907 97.066 .933 97.999 .676 98.675 .410 99.084 .326 99.411 .260 99.671 .145 99.816 .133 99.949 .051 100.000
Extraction Sums of Squared Loadings Total % of Variance Cumulative % 8.658 61.841 61.841 2.597 18.550 80.391
Extraction Method: Principal Component Analysis.
113
Rotation Sums of Squared Loadings Total % of Variance Cumulative % 8.458 60.413 60.413 2.797 19.978 80.391
Scree Plot
10
Eigenvalue
8
6
4
2
0 1
2
3
4
5
6
7
8
9
10
Component Number
Component Matrixa
1 Infant mortality (deaths per 1000 live births) Average female life expectancy People who read (%) Birth rate per 1000 people Females who read (%) Fertility: average number of kids Average male life expectancy Males who read (%) People living in cities (%) Death rate per 1000 people Gross domestic product / capita Birth to death ratio Population increase (% per year)) cropgrow
Component 2
-.951
-.168
.951
.222
.934 -.932 .923
-.064 .253 -.099
-.916
.235
.903
.315
.893 .741
-.108 .350
-.657
-.638
.611
.001
-.088
.927
-.650
.711
.150
-.619
Extraction Method: Principal Component Analysis. a. 2 components extracted.
114
11
12
13
14
Rotated Component Matrixa 1 Average female life expectancy Infant mortality (deaths per 1000 live births) Average male life expectancy People who read (%) Females who read (%) Birth rate per 1000 people Males who read (%) Fertility: average number of kids People living in cities (%) Death rate per 1000 people Gross domestic product / capita Birth to death ratio Population increase (% per year)) cropgrow
Component 2
.975
.046
-.965
.008
.945
.145
.907 .889 -.870 .859
-.232 -.265 .418 -.268
-.858
.397
.792
.209
-.762
-.508
.601
-.110
.082
.927
-.510
.817
.035
-.636
Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 3 iterations.
Component Transformation Matrix Component 1 2
1 .983 .182
2 -.182 .983
Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.
Component Plot in Rotated Space
b_to_d
1.0 pop_incr
0.5
birth_rt
Component 2
fertilty
urban
lifeexpm
babymort 0.0
gdp_cap
lifeexpf
lit_male
literacy lit_fema
death_rt
-0.5
cropgrow
-1.0 -1.0
-0.5
0.0
Component 1
115
0.5
1.0
Component Score Coefficient Matrix
1 People living in cities (%) Average female life expectancy Average male life expectancy People who read (%) Population increase (% per year)) Infant mortality (deaths per 1000 live births) Gross domestic product / capita Birth rate per 1000 people Death rate per 1000 people Birth to death ratio Fertility: average number of kids cropgrow Males who read (%) Females who read (%)
Component 2 .109 .117 .124
.064
.125
.100
.102
-.044
-.024
.283
-.120
-.044
.069
-.013
-.088
.115
-.119
-.228
.055
.353
-.088
.108
-.026 .094 .098
-.238 -.059 -.057
Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. Component Scores.
Here are the answers to the questions: 1. Although there were 109 countries in the data set, due to missing data in some of the variables, there were 84 countries used in the analysis. This number is obtained from the table of descriptive statistics in the column labeled N. 2. The KMO statistic has the value .87. Values above .50 are considered acceptable; a value above .80 is considered meritorious. 3. The variables with the lowest communalities were gross domestic product/ capita (.373) and cropgrow (.406). 4. Two principal components were retained. 5. Principal component 1 had an eigenvalue of 8.66 and principal component 2 had an eigenvalue of 2.60. 6. The two retained principal components accounted for about .80 of the total variance. 7. The three variables loading highest on each component are shown in the table below: Component 1 Loading Component 2 Loading lifeexpf .98 b_to_d .93 babymort -.97 pop_incr .82 lifeexpm ,95 cropgrow -.64 8. Component 1 may be how developed a country is in terms of its infrastructure and ability to provide healthcare to its inhabitants. Component 2 may be rate of development and growth.
116
Exercise 18.3 These are the screens that were used to analyze the data:
117
Here is the output produced:
118
Descriptive Statistics Mean
Std. Deviation
Analysis N
q1
1.98
1.086
84
q2
2.48
1.246
84
q3
4.01
1.035
84
q4
2.30
1.220
84
q5
2.43
1.254
84
q6
3.42
1.282
84
q7
3.80
1.062
84
q8
2.62
1.270
84
q9
2.27
1.274
84
q10
1.42
.853
84
q11
2.13
1.315
84
q12
1.95
1.171
84
q13
2.69
1.481
84
q14
3.75
.903
84
q15
2.14
1.214
84
Correlation Matrix q1
q2
q3
q4
q5
q6
q7
q8
q9
q10
q11
q12
q13
q14
q15
Correlation q1
1.000
.409
-.396
.424
.229
-.097
-.119
.343
.405
.284
.112
.520
.085
-.043
-.116
q2
.409
1.000
-.135
.341
.230
-.178
-.099
.580
.334
.264
-.097
.585
-.076
-.075
-.300
q3
-.396
-.135
1.000
-.003
-.152
.450
.287
-.235
-.240
.035
-.072
-.099
.105
.312
-.136
q4
.424
.341
-.003
1.000
.514
-.180
-.083
.510
.652
.053
.103
.415
.112
-.008
.003
q5
.229
.230
-.152
.514
1.000
-.225
-.169
.626
.808
.034
.104
.375
.066
-.298
.070
q6
-.097
-.178
.450
-.180
-.225
1.000
.399
-.308
-.226
.005
-.004
-.123
.069
.476
-.008
q7
-.119
-.099
.287
-.083
-.169
.399
1.000
-.344
-.163
-.265
-.145
-.182
-.033
.524
-.099
q8
.343
.580
-.235
.510
.626
-.308
-.344
1.000
.654
.093
.102
.595
.033
-.221
-.019
q9
.405
.334
-.240
.652
.808
-.226
-.163
.654
1.000
.082
.137
.388
.192
-.233
.029
q10
.284
.264
.035
.053
.034
.005
-.265
.093
.082
1.000
.187
.334
.189
-.113
.105
q11
.112
-.097
-.072
.103
.104
-.004
-.145
.102
.137
.187
1.000
.208
.392
-.043
.645
q12
.520
.585
-.099
.415
.375
-.123
-.182
.595
.388
.334
.208
1.000
.033
-.011
-.122
q13
.085
-.076
.105
.112
.066
.069
-.033
.033
.192
.189
.392
.033
1.000
.005
.394
q14
-.043
-.075
.312
-.008
-.298
.476
.524
-.221
-.233
-.113
-.043
-.011
.005
1.000
-.077
q15
-.116
-.300
-.136
.003
.070
-.008
-.099
-.019
.029
.105
.645
-.122
.394
-.077
1.000
KMO and Bartlett's Test Kaiser-Meyer-Olkin Measure of Sampling Adequacy. Bartlett's Test of Sphericity Approx. Chi-Square
.666 553.949
df
105
Sig.
.000
119
Communalities Initial q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 q15
Extraction
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
.754 .685 .877 .653 .799 .613 .702 .735 .840 .735 .731 .721 .532 .728 .786
Extraction Method: Principal Component Analysis. Total Variance Explained Comp onent
Initial Eigenvalues Total
Extraction Sums of Squared Loadings
% of Variance Cumulative %
Total
% of Variance Cumulative %
Rotation Sums of Squared Loadings Total
% of Variance Cumulative %
1
4.372
29.144
29.144
4.372
29.144
29.144
3.168
21.117
21.117
2
2.137
14.248
43.392
2.137
14.248
43.392
2.155
14.365
35.482
3
1.925
12.834
56.225
1.925
12.834
56.225
2.111
14.076
49.558
4
1.424
9.492
65.718
1.424
9.492
65.718
2.078
13.852
63.410
5
1.034
6.895
72.613
1.034
6.895
72.613
1.380
9.203
72.613
6
.757
5.050
77.663
7
.650
4.335
81.997
8
.575
3.833
85.830
9
.525
3.501
89.332
10
.442
2.950
92.281
11
.396
2.643
94.924
12
.267
1.781
96.705
13
.205
1.364
98.069
14
.173
1.151
99.220
15
.117
.780
100.000
Extraction Method: Principal Component Analysis.
120
Component Matrixa Component 1 q8 q9 q5 q12 q4 q1 q2 q15 q11 q13 q14 q6 q7 q10 q3
2 .834 .806 .725 .697 .667 .609 .608 .020 .213 .125 -.379 -.444 -.433 .290 -.395
3 -.116 .009 .025 -.198 -.118 -.144 -.446 .856 .742 .570 -.280 -.112 -.309 .179 -.151
4 .021 .146 .055 .295 .319 .179 .098 .155 .320 .425 .647 .631 .497 .195 .512
Extraction Method: Principal Component Analysis. a. 5 components extracted.
121
5 -.120 -.410 -.490 .330 -.304 .330 .324 -.091 .104 .009 -.004 .039 -.276 .694 -.089
.107 .024 .172 -.002 .000 -.470 .042 -.141 -.149 .105 -.295 .062 -.311 .316 .655
Rotated Component Matrixa Component 1 q9 q5 q4 q8 q10 q12 q2 q1 q15 q11 q13 q14 q7 q6 q3
2 .888 .864 .774 .735 -.135 .431 .357 .287 -.019 .065 .123 -.129 -.043 -.196 -.052
3 .100 -.019 .196 .339 .743 .722 .669 .589 -.201 .118 .092 .046 -.238 .044 -.006
4 .111 .068 .057 -.044 .266 .005 -.309 .042 .852 .836 .687 -.013 -.112 .102 -.032
Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 6 iterations.
122
5 -.121 -.218 .098 -.257 -.226 .013 -.045 .125 -.098 -.021 .055 .840 .794 .645 .326
-.118 .013 -.051 -.112 .210 -.117 -.110 -.554 -.097 -.116 .183 .056 .030 .383 .876
Component Score Coefficient Matrix Component 1 q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 q15
2 -.041 .015 .096 .286 .348 -.006 .088 .213 .328 -.194 -.026 .033 .043 .004 -.005
3 .249 .308 .065 -.035 -.173 .091 -.121 .047 -.117 .462 .048 .325 .043 .060 -.118
4 .034 -.153 -.025 .017 .007 .067 -.024 -.042 .034 .116 .401 -.002 .323 .027 .406
5 .200 .014 .011 .122 -.060 .277 .427 -.069 .023 -.159 .047 .065 .032 .448 -.007
-.434 .002 .689 .031 .141 .188 -.141 .053 .018 .249 -.104 -.012 .148 -.120 -.103
Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.
Component Score Covariance Matrix Compo nent 1 2 3 4 5
1 1.000 .000 .000 .000 .000
2
3
.000 1.000 .000 .000 .000
4
.000 .000 1.000 .000 .000
.000 .000 .000 1.000 .000
5 .000 .000 .000 .000 1.000
Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.
Of most importance to the question being addressed in this problem is the data shown in the Rotated Component Matrix table. The values in the table are loadings of the individual questions on the five retained components. Component 1 corresponds most strongly to questions 9, 5, 4, and 8. The text of these questions is shown below: 9. Some boys/girls push and shove other kids. How often do you do this? 5. Some boys/girls hit other kids. How often do you do this? How often do you do this? 4. When they are mad at someone, some boys/girls get back at the kid by not letting the kid be in their group anymore. How often do you do this? 8. Some boys/girls yell at other and call them mean names. How often 123
do you do this? These four questions all seem to have aggression in common. Component 2 corresponds most strongly to questions 10, 12, 2, and 1. The text of these questions is shown below: 10. Some boys/girls tell their friends that they will stop liking them unless the friends do what they say. How often do you tell friends this? 12. Some boys/girls try to keep other from liking a kid by saying mean things about the kid. How often do you do this? 2. Some boys/girls try to keep certain people from being in their group when it is time to play or do an activity. How often do you do this? 1. Some boys/girls tell lies (fibs/make up stories) about a kid so that others won’t like that kid anymore. How often do you do this? These questions all appear to have relational aggression in common. Component 3 corresponds most strongly to questions 15, 11, and 12. The text of these questions is shown below: 15. Some boys/girls have a lot of kids who like to play with them. How often do the other kids like to play with you? 11. Some boys/girls have a lot of friends. How often do you have a lot of friends? 13. Some boys/girls wish that they had more friends. How often do you feel this way? These questions appear to deal with social connections. Component 4 corresponds most strongly to questions 14, 7, and 6. The text of these questions is shown below: 14. Some boys/girls say or do nice things for other kids. How often do you do this? 7. Some boys/girls help out other kids when they need it .How often do you do this? 6. Some boys/girls let other know that they care about them. How often do you do this? These questions appear to have positive social behavior in common. Component 5 corresponds most strongly just to question 3. The text of the question is shown below: 3. Some boys/girls try to cheer up other kids who feel upset or sad. The question appears to also have positive social behavior, and seems to logically belong to component 4. However, it may differ from the questions comprising component 4 in that it may have an element of humor in it.
124
Chapter 19 Exercise 19.1 These are the screens that show the requests made for the analysis:
125
Here is the output produced: Group Statistics Valid N (listwise) own 1.00
Mean iq yrsed
2.00
iq iq iq
Weighted
95.0000
4.08248
4
4.000
11.7500
2.06155
4
4.000
4.08248
4
4.000
17.0000
2.58199
4
4.000
103.2500
4.57347
4
4.000
23.0000
4.24264
4
4.000
101.0833
5.96137
12
12.000
17.2500
5.56164
12
12.000
yrsed Total
Unweighted
105.0000
yrsed 3.00
Std. Deviation
yrsed
Tests of Equality of Group Means Wilks' Lambda iq yrsed
F
.416 .255
df1
6.309 13.150
df2 2 2
9 9
Covariance Matrices own 1.00
iq iq
2.00
iq
3.333
3.333
4.250
16.667
6.667
yrsed 3.00
yrsed
16.667
yrsed
6.667
6.667
iq
20.917
18.000
yrsed
18.000
18.000
126
Sig. .019 .002
Analysis 1 Box's Test of Equality of Covariance Matrices Log Determinants own
Rank
Log Determinant
1.00 2.00 3.00 Pooled within-groups
2 2 2 2
4.090 4.200 3.961 4.468
The ranks and natural logarithms of determinants printed are those of the group covariance matrices.
Test Results Box's M F Approx.
3.462 .390
df1
6
df2
2018.769
Sig.
.886
Tests null hypothesis of equal population covariance matrices.
Summary of Canonical Discriminant Functions Eigenvalues Functio n
Eigenvalue
% of Variance
3.259a 1.109a
1 2
Canonical Correlation
Cumulative %
74.6 25.4
74.6 100.0
.875 .725
a. First 2 canonical discriminant functions were used in the analysis.
Wilks' Lambda Test of Function(s)
Wilks' Lambda
1 through 2 2
Chi-square
.111 .474
df
18.661 6.343
Standardized Canonical Discriminant Function Coefficients Function 1 iq yrsed
-.560 1.314
127
2 1.298 -.522
Sig. 4 1
.001 .012
Structure Matrix Function 1
2 *
yrsed iq
.918 .369
.396 .929*
Pooled within-groups correlations between discriminating variables and standardized canonical discriminant functions Variables ordered by absolute size of correlation within function. *. Largest absolute correlation between each variable and any discriminant function
Functions at Group Centroids Function own
1
1.00 2.00 3.00
-1.527 -.622 2.148
2 -.933 1.238 -.305
Unstandardized canonical discriminant functions evaluated at group means
Classification Statistics Classification Processing Summary Processed Excluded
12 0
Missing or out-of-range group codes At least one missing discriminating variable Used in Output
0 12
Prior Probabilities for Groups Cases Used in Analysis own 1.00 2.00 3.00 Total
Prior
Unweighted
.333 .333 .333 1.000
4 4 4 12
128
Weighted 4.000 4.000 4.000 12.000
Classification Function Coefficients own
iq yrsed (Constant)
1.00
2.00
3.00
9.244 -7.732 -394.775
9.788 -7.714 -449.389
8.952 -6.282 -391.004
These are the functions used to answer question 1.
Fisher's linear discriminant functions Classification Resultsb,c Predicted Group Membership own Original
Count
%
Cross-validateda
Count
%
1.00
2.00
3.00
Total
1.00
4
0
0
4
2.00
0
4
0
4
3.00
0
0
4
4
1.00
100.0
.0
.0
100.0
2.00
.0
100.0
.0
100.0
3.00
.0
.0
100.0
100.0
1.00
3
1
0
4
2.00
1
3
0
4
3.00
0
0
4
4
1.00
75.0
25.0
.0
100.0
2.00
25.0
75.0
.0
100.0
3.00
.0
.0
100.0
100.0
a. Cross validation is done only for those cases in the analysis. In cross validation, each case is classified by the functions derived from all cases other than that case. b. 100.0% of original grouped cases correctly classified. c. 83.3% of cross-validated grouped cases correctly classified.
Answers: 1. The Fisher classification functions are G1 = 9.244 × IQ − 7.732 × YrsEd − 394.775 G2 = 9.788 × IQ − 7.714 × YrsEd − 449.389 G3 = 8.952 × IQ − 6.282 × YrsEd − 391.004 2. 3.
Based on the leave-one-out procedure, 83.3% of the cases will be correctly classified. For case 1: G1 = 9.244 × 109 − 7.732 × 26 − 394.775 = 411.78 G2 = 9.788 × 109 − 7.714 × 26 − 449.389 = 416.94 G3 = 8.952 × 109 − 6.282 × 26 − 391.004 = 421.43 The largest of the 3 values corresponds to G3, so the case is classified as category 3, a renter. 129
For case 2 G1 = 9.244 × 99 − 7.732 × 12 − 394.775 = 427.60 G2 = 9.788 × 99 − 7.714 × 12 − 449.389 = 427.06 G3 = 8.952 × 99 − 6.282 × 12 − 391.004 = 419.86 The largest of the 3 values corresponds to G1, so the case is classified as category 1, a non-owner.
Exercise 19.3 These are the screens that show the requests made for the analysis:
130
Selected output is shown below: Analysis Case Processing Summary Unweighted Cases Valid Excluded
N
Missing or out-of-range group codes At least one missing discriminating variable Both missing or out-of-range group codes and at least one missing discriminating variable Total
Total
131
Percent 507 0
100.0 .0
0
.0
0
.0
0 507
.0 100.0
Group Statistics Gender; 1 for males and 0 for females 0 Mean Biacromial diameter (in cm) Bitrochanteric diameter (in cm) Elbow diameter (in cm) Wrist diameter (in cm) Biiliac diameter, or pelvic breadth (in cm) Chest depth (in cm) Chest diameter (in cm) Knee diameter (in cm) Ankle diameter (in cm) Shoulder girth (in cm) Chest girth (in cm) Waist girth (in cm) Navel girth (in cm) Hip girth (in cm) Thigh girth (in cm) Bicep girth (in cm) Forearm girth (in cm) Knee girth (in cm) Calf girth (in cm) Ankle girth (in cm) Wrist girth (in cm) Weight (in kg) Height (in cm)
Std. Deviation
male Valid N (listwise) Unweighted Weighted
Mean
Std. Deviation
Valid N (listwise) Unweighted Weighted
36.5031
1.77922
260
260.000
41.2413
2.08716
247
247.000
31.4615
2.04918
260
260.000
32.5267
1.86513
247
247.000
12.3669 9.8742
.83637 .66163
260 260
260.000 260.000
14.4571 11.2462
.88254 .63590
247 247
247.000 247.000
27.5815
2.30748
260
260.000
28.0915
2.06710
247
247.000
17.7246 26.0973 18.0969 13.0265 100.3038 86.0600 69.8035 83.7458 95.6527 57.1958 28.0973 23.7604 35.2600 35.0062 21.2058 15.0592 60.6004 164.8723
1.83206 1.81881 1.18660 .86606 6.47060 6.17041 7.58775 9.96163 6.94073 4.63600 2.70948 1.68225 2.57808 2.61313 1.43882 .84941 9.61570 6.54460
260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260
260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000 260.000
20.8065 29.9490 19.5619 14.7441 116.5016 100.9899 84.5332 87.6623 97.7632 56.4980 34.4036 28.2405 37.1955 37.2069 23.1591 17.1903 78.1445 177.7453
2.14363 2.08311 1.07136 .94424 6.49802 7.20902 8.78224 8.38488 6.22804 4.24667 2.98204 1.77932 2.27300 2.64514 1.72909 .90800 10.51289 7.18363
247 247 247 247 247 247 247 247 247 247 247 247 247 247 247 247 247 247
247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000 247.000
Tests of Equality of Group Means Wilks' Lambda
F
df1
df2
Sig.
Biacromial diameter (in cm)
.399
759.223
1
505
.000
Biiliac diameter, or pelvic breadth (in cm)
.987
6.845
1
505
.009
Bitrochanteric diameter (in cm)
.931
37.347
1
505
.000
Chest depth (in cm)
.624
303.814
1
505
.000
Chest diameter (in cm)
.506
493.162
1
505
.000
Elbow diameter (in cm)
.403
749.653
1
505
.000
Wrist diameter (in cm)
.472
565.636
1
505
.000
Knee diameter (in cm)
.704
212.183
1
505
.000
Ankle diameter (in cm)
.525
456.265
1
505
.000
Shoulder girth (in cm)
.390
790.480
1
505
.000
Chest girth (in cm)
.445
629.622
1
505
.000
Waist girth (in cm)
.552
409.577
1
505
.000
Navel girth (in cm)
.957
22.821
1
505
.000
Hip girth (in cm)
.975
12.939
1
505
.000
Thigh girth (in cm)
.994
3.114
1
505
.078
Bicep girth (in cm)
.448
622.151
1
505
.000
Forearm girth (in cm)
.373
849.251
1
505
.000
Knee girth (in cm)
.863
80.083
1
505
.000
Calf girth (in cm)
.850
88.775
1
505
.000
Ankle girth (in cm)
.725
191.928
1
505
.000
Wrist girth (in cm)
.404
745.468
1
505
.000
Weight (in kg)
.567
385.030
1
505
.000
Height (in cm)
.531
445.610
1
505
.000
132
Analysis 1 Box's Test of Equality of Covariance Matrices Log Determinants Gender; 1 for males and 0 for females 0 male Pooled within-groups
Rank 23 23 23
Log Determinant 16.971 21.598 20.724
The ranks and natural logarithms of determinants printed are those of the group covariance matrices.
Test Results Box's M F
Approx. df1 df2 Sig.
756.730 2.613 276 771799.8 .000
Tests null hypothesis of equal population covariance matrices.
Summary of Canonical Discriminant Functions Eigenvalues Function 1
Eigenvalue % of Variance 7.128a 100.0
Canonical Correlation .936
Cumulative % 100.0
a. First 1 canonical discriminant functions were used in the analysis.
Wilks' Lambda Test of Function(s) 1
Wilks' Lambda .123
Chi-square 1034.041
133
df 23
Sig. .000
Standardized Canonical Discriminant Function Coefficients
Structure Matrix
Function 1 Biacromial diameter (in cm) Bitrochanteric diameter (in cm) Elbow diameter (in cm) Wrist diameter (in cm) Biiliac diameter, or pelvic breadth (in cm) Chest depth (in cm) Chest diameter (in cm) Knee diameter (in cm) Ankle diameter (in cm) Shoulder girth (in cm) Chest girth (in cm) Waist girth (in cm) Navel girth (in cm) Hip girth (in cm) Thigh girth (in cm) Bicep girth (in cm) Forearm girth (in cm) Knee girth (in cm) Calf girth (in cm) Ankle girth (in cm) Wrist girth (in cm) Weight (in kg) Height (in cm)
Forearm girth (in cm) Shoulder girth (in cm) Biacromial diameter (in cm) Elbow diameter (in cm) Wrist girth (in cm) Chest girth (in cm) Bicep girth (in cm) Wrist diameter (in cm) Chest diameter (in cm) Ankle diameter (in cm) Height (in cm) Waist girth (in cm) Weight (in kg) Chest depth (in cm) Knee diameter (in cm) Ankle girth (in cm) Calf girth (in cm) Knee girth (in cm) Bitrochanteric diameter (in cm) Navel girth (in cm) Hip girth (in cm) Biiliac diameter, or pelvic breadth (in cm) Thigh girth (in cm)
.293 -.100 .153 -.052 -.109 .134 -.061 .093 .246 .172 -.165 1.335 -.746 -.237 -.502 .424 .506 -.078 -.039 .005 -.093 -.633 .414
134
Function 1 .486 .469 .459 .456 .455 .418 .416 .396 .370 .356 .352 .337 .327 .291 .243 .231 .157 .149 .102 .080 .060 .044 -.029
Canonical Discriminant Function Coefficients Function 1 Biacromial diameter (in cm) Bitrochanteric diameter (in cm) Elbow diameter (in cm) Wrist diameter (in cm) Biiliac diameter, or pelvic breadth (in cm) Chest depth (in cm) Chest diameter (in cm) Knee diameter (in cm) Ankle diameter (in cm) Shoulder girth (in cm) Chest girth (in cm) Waist girth (in cm) Navel girth (in cm) Hip girth (in cm) Thigh girth (in cm) Bicep girth (in cm) Forearm girth (in cm) Knee girth (in cm) Calf girth (in cm) Ankle girth (in cm) Wrist girth (in cm) Weight (in kg) Height (in cm) (Constant)
.151 -.051 .178 -.080 -.050 .067 -.031 .082 .272 .027 -.025 .163 -.081 -.036 -.113 .149 .292 -.032 -.015 .003 -.106 -.063 .060 -21.313
Unstandardized coefficients
Functions at Group Centroids Gender; 1 for males and 0 for females 0 male
Function 1 -2.597 2.734
Unstandardized canonical discriminant functions evaluated at group means
Classification Statistics Prior Probabilities for Groups
Gender; 1 for males and 0 for females 0 male Total
Prior .500 .500 1.000
135
Cases Used in Analysis Unweighted Weighted 260 260.000 247 247.000 507 507.000
Classification Function Coefficients Gender; 1 for males and 0 for females 0 male Biacromial diameter (in cm) Bitrochanteric diameter (in cm) Elbow diameter (in cm) Wrist diameter (in cm) Biiliac diameter, or pelvic breadth (in cm) Chest depth (in cm) Chest diameter (in cm) Knee diameter (in cm) Ankle diameter (in cm) Shoulder girth (in cm) Chest girth (in cm) Waist girth (in cm) Navel girth (in cm) Hip girth (in cm) Thigh girth (in cm) Bicep girth (in cm) Forearm girth (in cm) Knee girth (in cm) Calf girth (in cm) Ankle girth (in cm) Wrist girth (in cm) Weight (in kg) Height (in cm) (Constant)
2.868
3.675
-2.550
-2.823
-2.398 6.674
-1.449 6.248
3.074
2.810
6.570 2.548 14.742 4.709 3.124 5.147 9.361 -1.247 7.220 9.264 -.947 17.273 3.708 9.740 .004 -.797 -27.972 11.390 -2105.098
6.929 2.381 15.181 6.159 3.266 5.015 10.230 -1.678 7.029 8.663 -.152 18.831 3.538 9.660 .022 -1.361 -28.307 11.712 -2219.080
Fisher's linear discriminant functions
Classification Resultsb,c Gender; 1 for males and 0 for females Original
Count
0 female 1 male
% a
Cross-validated
Count
0 female
0 female
1 male
Total
257
3
260
5
242
247
98.8
1.2
100.0
1 male
2.0
98.0
100.0
0 female
257
3
260
1 male %
Predicted Group Membership
0 female 1 male
6
241
247
98.8
1.2
100.0
2.4
97.6
100.0
a. Cross validation is done only for those cases in the analysis. In cross validation, each case is classified by the functions derived from all cases other than that case. b. 98.4% of original grouped cases correctly classified. c. 98.2% of cross-validated grouped cases correctly classified.
136
Answers: 1. Yes. The function has a Wilks’ lambda = .12, χ2 (23, N = 507) = 1,034.04, p < .001. 2. The discriminant function successfully predicted computer gender of 98% of the cases in the sample. 3. The two variables that were most strongly correlated with the discriminant function were forearm girth and shoulder girth. Based on the standardized dircriminant function coefficients, waist girth and navel girth made the greatest independent contributions to discriminating between males and females. To answer question 4, a discriminant analysis was conducted using the predictors forearm girth, shoulder girth, waist girth and navel girth. Selected output is shown below: Analysis Case Processing Summary Unweighted Cases Valid Excluded Missing or out-of-range group codes At least one missing discriminating variable Both missing or out-of-range group codes and at least one missing discriminating variable Total Total
N 507
Percent 100.0
0
.0
0
.0
0
.0
0 507
.0 100.0
Group Statistics Gender; 1 for males and 0 for females 0
male
Total
Forearm girth (in cm) Shoulder girth (in cm) Waist girth (in cm) Navel girth (in cm) Forearm girth (in cm) Shoulder girth (in cm) Waist girth (in cm) Navel girth (in cm) Forearm girth (in cm) Shoulder girth (in cm) Waist girth (in cm) Navel girth (in cm)
Mean 23.7604 100.3038 69.8035 83.7458 28.2405 116.5016 84.5332 87.6623 25.9430 108.1951 76.9795 85.6538
137
Std. Deviation 1.68225 6.47060 7.58775 9.96163 1.77932 6.49802 8.78224 8.38488 2.83058 10.37483 11.01269 9.42413
Valid N (listwise) Unweighted Weighted 260 260.000 260 260.000 260 260.000 260 260.000 247 247.000 247 247.000 247 247.000 247 247.000 507 507.000 507 507.000 507 507.000 507 507.000
Tests of Equality of Group Means
Forearm girth (in cm) Shoulder girth (in cm) Waist girth (in cm) Navel girth (in cm)
Wilks' Lambda .373 .390 .552 .957
F 849.251 790.480 409.577 22.821
df1 1 1 1 1
df2 505 505 505 505
Sig. .000 .000 .000 .000
Covariance Matrices Gender; 1 for males and 0 for females 0
male
Forearm girth (in cm) Shoulder girth (in cm) Waist girth (in cm) Navel girth (in cm) Forearm girth (in cm) Shoulder girth (in cm) Waist girth (in cm) Navel girth (in cm)
Forearm girth (in cm) 2.830 8.134 9.034 10.718 3.166 8.132 6.573 6.359
Shoulder girth (in cm) 8.134 41.869 35.636 39.538 8.132 42.224 33.152 29.374
Waist girth (in cm) 9.034 35.636 57.574 63.132 6.573 33.152 77.128 64.952
Box's Test of Equality of Covariance Matrices Log Determinants Gender; 1 for males and 0 for females 0 male Pooled within-groups
Rank 4 4 4
Log Determinant 10.506 10.877 10.880
The ranks and natural logarithms of determinants printed are those of the group covariance matrices.
Test Results Box's M F
Approx. df1 df2 Sig.
97.835 9.700 10 1211974 .000
Tests null hypothesis of equal population covariance matrices.
138
Navel girth (in cm) 10.718 39.538 63.132 99.234 6.359 29.374 64.952 70.306
Summary of Canonical Discriminant Functions Eigenvalues Function 1
Eigenvalue % of Variance 3.309a 100.0
Canonical Correlation .876
Cumulative % 100.0
a. First 1 canonical discriminant functions were used in the analysis.
Wilks' Lambda Test of Function(s) 1
Wilks' Lambda .232
Chi-square 734.696
df 4
Sig. .000
Standardized Canonical Discriminant Function Coefficients Function 1 .588 .324 1.019 -1.249
Forearm girth (in cm) Shoulder girth (in cm) Waist girth (in cm) Navel girth (in cm) Structure Matrix
Forearm girth (in cm) Shoulder girth (in cm) Waist girth (in cm) Navel girth (in cm)
Function 1 .713 .688 .495 .117
Pooled within-groups correlations between discriminating variables and standardized canonical discriminant functions Variables ordered by absolute size of correlation within function. Canonical Discriminant Function Coefficients
Forearm girth (in cm) Shoulder girth (in cm) Waist girth (in cm) Navel girth (in cm) (Constant)
Function 1 .340 .050 .124 -.135 -12.190
Unstandardized coefficients
139
Functions at Group Centroids Function 1 -1.769 1.863
Gender; 1 for males and 0 for females 0 male
Unstandardized canonical discriminant functions evaluated at group means
Classification Statistics Classification Function Coefficients
Forearm girth (in cm) Shoulder girth (in cm) Waist girth (in cm) Navel girth (in cm) (Constant)
Gender; 1 for males and 0 for females 0 male 3.168 4.401 2.081 2.263 -.793 -.341 .415 -.077 -132.390 -176.832
Fisher's linear discriminant functions
Classification Resultsb,c
Original
Count %
Cross-validated a
Count %
Gender; 1 for males and 0 for females 0 male 0 male 0 male 0 male
Predicted Group Membership 0 male 249 11 10 237 95.8 4.2 4.0 96.0 249 11 11 236 95.8 4.2 4.5 95.5
Total 260 247 100.0 100.0 260 247 100.0 100.0
a. Cross validation is done only for those cases in the analysis. In cross validation, each case is classified by the functions derived from all cases other than that case. b. 95.9% of original grouped cases correctly classified. c. 95.7% of cross-validated grouped cases correctly classified.
4.
Yes. The function has a Wilks’ lambda = .23, χ2 (4, N = 507) = 734.70, p < .001. The discriminant function successfully predicted computer gender of 96% of the cases in the sample. The Fisher functions could be used to predict a person’s gender based on values of the four variables. The functions are: Gender female = 3.168 forearm + 2.081shoulder − .793waist + .415navel − 132.39 Gendermale = 4.401 forearm + 2.263shoulder − .341waist − .077navel − 176.832 . A person is given the gender classification of the function that has the higher value.
140
![Pearson Student Mobile Device Survey [PDF]](https://m.moam.info/img/260x300/pearson-student-mobile-device-survey-pdf_6495f353098a9e55498b4623.jpg)
![Pearson Student Mobile Device Survey [PDF]](https://m.moam.info/img/260x300/pearson-student-mobile-device-survey-pdf_647d8613098a9ede188b4666.jpg)
![Pearson Student Mobile Device Survey 2013 [PDF]](https://m.moam.info/img/260x300/pearson-student-mobile-device-survey-2013-pdf_6495f350098a9e2d388b462f.jpg)
![Pearson Student Mobile Device Survey 2015 [PDF]](https://m.moam.info/img/260x300/pearson-student-mobile-device-survey-2015-pdf_647dd207098a9e89398b45b6.jpg)
![Student Mobile Device Survey - Pearson [PDF]](https://m.moam.info/img/260x300/student-mobile-device-survey-pearson-pdf_647d7546098a9e3a168b4585.jpg)
