Tables and Formulas for Moore, The Basic Practice of Statistics

Tables and Formulas for Moore, The Basic Practice of Statistics Exploring Data: Distributions • Remember overall pattern (shape, center, spread) and deviations (outliers).

• Least square regression line (in slope – intercept form): y = mx + b

• Mean (use a calculator): x1 + x2 + . . . + xn 1 x¯ = = Σ n n xi

sy slope = r s x

intercept = y¯ - (slope)( x¯ )

• Residuals: • Standard Deviation (use a calculator) 1 2 n - 1 Σ (xi - x¯ )

s= =

• Median: Arrange all observations from smallest to largest. The median M is located (n + 1)/2 observations from the beginning of the list. • Quartiles: The first quartile Q1 is the median of the observations whose position in the ordered list is to the left of the location of the overall median. The third quartile Q3 is the median of the observations to the right of the location of the overall median. • Five-number summary: Minimum, Q1, M, Q3, Maximum

residuals = observed y – predicted y = y - y^ Producing Data • Simple Random sample: Choose an SRS by giving every individual in the population a numerically label and using Table B of random digits to choose the sample.

Probability and Sampling Distributions • Probability Rules: • Any probability satisfies 0 ≤ P(A) ≤ 1. • The sample space S has probability P(S) = 1. • For any event A, P(A does not occur) = 1 – P(A). • If events A and B are disjoint, P(A or B) = P(A) + P(B).

• Standardized value of x: • Sampling Distribution of a sample mean x-μ σ Exploring Data: Relationships • Remember overall pattern (form, direction, strength) and deviations (outliers, influential observations). Z=

• Correlation (use a calculator): 1 xi - x¯ yi - y¯ r= n-1 Σ( s )( s ) x y

formulas for Moore.doc

• x¯ has mean μ and standard deviation

σ n

• x¯ has a Normal distribution if the population distribution is Normal. • Central Limit Theorem: ¯x is approximately Normal when n is large.

Basics of Inference • z confidence interval for a population mean (σ known, SRS from normal population): σ x¯ ± z* z* from N(0,1) n • Sample size for desired margin of error m: z* σ 2 n= m • z test statistic for H0: μ = μ0 (σ known, SRS from Normal population):

(

)

P-values from N(0,1)

Z* from N(0,1)

• z test statistic for H0: p = p0 (large SRS):

z=

^ p - p0 p0(1-p0) n

P-values from t(n-1)

• Two-sample t confidence interval for μ1 - μ2 (independent SRSs from Normal populations): S12 S22 + n1 n2 with conservative t* from t with df the smaller of n1-1 and n2 – 1. ( x¯1 - x¯2 ) ± t*

• Two-sample t test statistic for H0: μ1 = μ2 (independent SRSs from Normal populations): x¯1 - x¯2 S12 S22 + n1 n2

with conservative P-values from t with df the smaller of n1-1 and n2 – 1.

formulas for Moore.doc

P-values from N(0,1)

• Sample size for desired margin of error m: n=

• Matched pairs: To compare the responses to the two treatments, apply the one-sample t procedures to the observed differences.

t=

^ p (1- ^ p ) n

*

Inference About Means • t confidence interval for a population mean (SRS from Normal population): S x¯ ± t* t* from t(n-1) n • t test statistic for H0: μ = μ0 (SRS from Normal population): ¯x - μ0 S/ n

• Large-sample z confidence interval for p: ^ p ± z

¯x - μ0 z= σ/ n

t=

Inference About Proportions • Sampling distribution of a sample proportion: when the population is large and both np ≥ 10 and n(1-p) ≥10, ^ p is approximately Normal with mean p and standard deviation p (1-p) n

2 ( z* m ) p*(1-p*)

where p* is a guessed value for p or p* = .5 • Z confidence interval for p1 – p2 (large independent SRSs): (^ p1 - ^ p2 ) ± z* SE

Z* from N(0,1)

where the standard error of p^1 - p^2 is SE =

p2 (1-p ^ p1 (1-p ^1) ^ ^2) + n1 n2

• Two-sample z test statistic for H0: p1 = p2 (large independent SRSs):

z=

^ p1 - ^ p2 1 1 ^ p(1-p ^ )( + ) n1 n2

where ^ p is the pooled proportion of successes.