Confidence Intervals for One Mean

Confidence Intervals for One Mean General Confidence Interval Formula:

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Point Estimate  Margin of Error Point Estimate  (t-value)(Standard Error)

The point estimate is the best guess for the population mean:

The sample standard error is the best guess for the population standard error: The t-value should be found on the t-table by looking up t  n  1 . 2

Confidence Interval for One Population Mean  :

Under the assumptions that:

1. When the sample size is large relative to the population:

sample size  0.05 population size

Data come from a simple random sample or randomized experiment.

2. When the sample size is small relative to the population:

sample size  0.05, population size

Data come from a simple random sample or randomized experiment, and Data comes from an originally normal population.

Confidence Intervals for One Mean

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Example: A simple random sample of size n is drawn from the population of size N = 45,000. a) If the sample size is 12, what conditions must be satisfied to compute the confidence interval?

b) If the sample size is 2500, what conditions must be satisfied to compute the confidence interval?

c) To form a confidence interval for the population mean, does the population that the sample is drawn from always have to be approximately normal?


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Example . . . U.S. fire departments responded to an estimated average of 157,300 home structure fires involving cooking equipment per year. These fires caused an annual average of 380 civilian deaths, 4,920 civilian injuries, and $794 million in direct property damage. . . National estimates of reported fires were derived from the U.S. Fire Administration’s National Fire Incident Reporting System and NFPA’s annual fire department experience survey. [Ahrens]

[Ahrens]


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Example 1: Let us say that the National Fire Protection Association randomly sampled 1000 counties throughout the United States and found that U.S. fire departments responded to an estimated average of 157,300 home structure fires involving cooking equipment per year, with a standard deviation of 3,500 homes.

a) Based on this information, construct a 95% confidence interval for the mean number of times that the U.S. fire department responded to home structure fires involving cooking equipment.

b)

Construct a 99% confidence interval for the mean number of times that the U.S. fire department responded to home structure fires involving cooking equipment.

c) When we increased the level of confidence from 95% to 99%, what happened to the margin of error? What does this do to the confidence interval?

Confidence Intervals for One Mean d) Let us say that the National Fire Protection Association had instead randomly sampled 3000 counties throughout the United States (but still found the same mean and standard deviation). Form the 95% confidence interval.

e) What happens to the margin of error when the sample size increases?

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Example Continued: According to the Institute for Burn Medicine for San Diego and Imperial Counties, this is the correct method to extinguish a fire involving cooking equipment: http://www.youtube.com/watch?v=j95_gFWD9lc Below are the methods that the NFPA found residents used in attempting to extinguish the home structure fire involving cooking equipment.

Extinguishment Method Putting water on the fire Turning off power to appliance

Limitations of Method Water can cause a grease fire to spread An excellent first step but insufficient by itself unless the cooking equipment provides a fairly tight enclosure (e.g., oven, microwave oven) that will smother the fire without further action.

Smothering the fire

Using a lid to smother a pan fire is the preferred safe, effective way to extinguish a stovetop pan fire. The lid must be kept on until the pan has cooled or the fire could flare up again. . . Carrying burning material is clearly unsafe, and this action is even more dangerous if it involves carrying a pan with burning oil or grease or opening the door to an oven or microwave oven, which may result in a flare-up adding oxygen to the fire. This approach can be ineffective because it is dangerous to get close enough to the fire to apply materials like this and it is difficult to achieve full coverage sufficient to smother the fire. Also, some of these substances, such as flour, can be ignited. This approach can be effective but only if the right type of extinguisher . . . is used under the right conditions (e.g., not when the pressure of the stream could dislodge the pan and spread the fire, not when the fire is growing rapidly or threatening to cut off escape paths). Using a lid is safer.

Separating the burning material from heat source or moving outside Using flour, baking soda, salt, or other substances as an extinguishing agent Using a fire extinguisher

Blowing out the fire

Very small fires may be blown out, but this practice brings a person’s face (and possibly hair) close to flames.


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Example 2: The following data represent the direct property damage (in Millions of dollars) due to home fires involving cooking equipment, by year. The dollars for each year have been adjusted to their equivalent 2010 dollar amounts. [Data Source: Ahrens]

a) The population of interest is the direct property damage per year (in Millions of dollars) due to home fires involving cooking equipment from 1980 to 2010. What is the parameter that describes this population?

[Ahrens]


b) Ten years between 1980 and 2010 were randomly sampled. Here is the data, the boxplot, and the normal probability plot. Use this to construct a 99% confidence interval for the population mean cost of direct property damage per year due to fires involving cooking equipment.

d) Does the 99% confidence interval include the population parameter?

Year 1980 1983 1985 1987 1992 1995 1999 2000 2002 2006 Average

Damage $649 $750 $708 $761 $701 $637 $650 $653 $813 $739 $

Std Dev

$

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e) A second random sample of 10 days resulted in the following data. Construct another 99% confidence interval for the population mean cost of direct property damage per year due to fires involving cooking equipment. Year 1981 1985 1988 1991 1997 1998 2001 2005 2007 2010 Average Std Dev

f) Why are the confidence intervals for these two random samples different?

Damage $1832 $708 $850 $993 $767 $705 $641 $975 $568 $993 $903.2 $359.6


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g) The NFPA report states: “The previous data set contained the year 1981 with $1,832 Million dollars in property damage due to fires involving cooking equipment.” Let us remove 1981 from the data set. Now form the 99% confidence interval. What effect does the unusually large value have on the confidence interval? Year Damage 1981 $1832 1985 $708 1988 $850 1991 $993 1997 $767 1998 $705 2001 $641 2005 $975 2007 $568 2010 $993 Average $800 Std Dev $160.2


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Guidelines for Interpreting Confidence Intervals 1. Confidence intervals ARE statements about the POPULATION MEAN. 2. Confidence intervals SHOULD imply that the INTERVAL CAN VARY BASED ON THE SAMPLE DATA. 3. Confidence intervals ARE NOT statements about the SAMPLE MEAN. 4. Confidence intervals ARE NOT statements about INDIVIDUALS.

Example: The 95% confidence interval for the mean number of civilian deaths per year due to home fires involving cooking equipment had a lower bound of 406.5 deaths per year and an upper bound of 410.6 deaths per year. Below are four possible interpretations. Match each interpretation with the guideline that explains why the interpretation is correct or flawed.

a) There is a 95% probability the mean number of civilian deaths per year due to home fires involving cooking equipment was between 406.5 deaths per year and 410.6 deaths per year.

b) 95% of home fires involving cooking equipment will result in between 406.5 civilian deaths per year and 410.6 civilian deaths per year.

c) We are 95% confident that the mean number of civilian deaths due to home fires involving cooking equipment for the years sampled was between 406.5 deaths per year and 410.6 deaths per year.

d) We are 95% confident that the mean number of civilian deaths per year due to home fires involving cooking equipment was between 406.5 deaths per year and 410.6 deaths per year.


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Works Cited Ahrens , Marty (2012). HOME FIRES INVOLVING COOKING EQUIPMENT. Retrieved from the National Fire Protection Association: http://www.nfpa.org/assets/files/pdf/os.cooking.pdf Institute for Burn Medicine for San Diego and Imperial Counties (Producer). (2008, December 16). Death in the Kitchen (Warning) [Video file]. Retrieved from http://www.youtube.com/watch?v=j95_gFWD9lc


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Steps For Determining the Sample Size Needed To Estimate a Population Mean 1. Plug in the sample standard deviation the Margin of Error given in the problem. M= t  n  1 2

s n

2. Plug in the Z-Score for the appropriate confidence level: Z = 1.645 for 90% confidence Z = 1.96 for 95% confidence Z = 2.575 for 99% confidence 3. Solve for n. 4. The answer must be an integer, so always round up if you get a decimal.


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Example: Mirena is an intrauterine contraceptive that delivers small amounts of hormone to the uterus. . . Mirena offers contraception that’s over 99% effective; in fact, it is one of the most effective forms of birth control. . . Mirena contains a progestin hormone called levonorgestrel that is often used in birth control pills. . . . The levonorgestrel in Mirena reduces the menstral flow. [Guide]

On average, a woman’s menstrual flow lasts 3 to 4 days. It may be shorter or longer depending on the woman. [Guide] Assume that initial studies indicate that s = 3.6 days. How many subjects are needed to estimate the average menses length for a woman using Mirena with 90% confidence within a 1 day margin of error?

Confidence Intervals for One Mean Works Cited A guide for those considering birth control with Mirena. (February 2011). Wayne: New Jersey: Bayer Health Care Pharmaceuticals Inc.

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