Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problem 7.2.28b

Chapter 7, Problem 7.2.28b

Mean Body Temperature Data Set 5 “Body Temperatures” in Appendix B includes 106 body temperatures of adults for Day 2 at 12 AM, and they vary from a low of 96.5F to a high of 99.6F. Find the minimum sample size required to estimate the mean body temperature of all adults. Assume that we want 98% confidence that the sample mean is within 0.1F of the population mean.

b. Assume that sigma=0.62F, based on the value of s=0.62F for the sample of 106 body temperatures.

Verified step by step guidance

Step 1: Identify the formula for determining the minimum sample size required to estimate the population mean. The formula is: n = (Z * σ / E)^2, where n is the sample size, Z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and E is the margin of error.

Step 2: Determine the values for the variables in the formula. From the problem, the confidence level is 98%, so the z-score (Z) corresponding to 98% confidence can be found using a z-table or statistical software. The population standard deviation (σ) is given as 0.62°F, and the margin of error (E) is 0.1°F.

Step 3: Substitute the values into the formula. Replace Z with the z-score for 98% confidence, σ with 0.62, and E with 0.1. The formula becomes: n = (Z * 0.62 / 0.1)^2.

Step 4: Simplify the expression inside the parentheses first. Multiply the z-score (Z) by 0.62, then divide the result by 0.1.

Step 5: Square the result from Step 4 to calculate the minimum sample size (n). Round up to the nearest whole number, as sample size must be an integer.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Size Determination

Sample size determination is a statistical process used to calculate the number of observations needed to achieve a desired level of precision in estimating a population parameter. In this context, it involves using the desired confidence level and margin of error to ensure that the sample mean accurately reflects the population mean within specified limits.

Confidence Interval

A confidence interval is a range of values, derived from a data set, that is likely to contain the population parameter with a specified level of confidence. For example, a 98% confidence interval means that if we were to take many samples and build intervals, approximately 98% of those intervals would contain the true population mean.

Standard Deviation and Population Variance

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this scenario, the standard deviation (sigma) of 0.62F indicates how much individual body temperatures deviate from the mean. This value is crucial for calculating the sample size needed to achieve the desired confidence level and margin of error.

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Textbook Question

Wiggle Your Ears Find the sample size needed to estimate the percentage of adults who can wiggle their ears. Use a margin of error of 3 percentage points and use a confidence level of 99%.

b. Assume that 22% of adults can wiggle their ears (based on data from Soul Publishing).

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Textbook Question

Smart Phone Apple is planning for the launch of a new and improved iPhone. The marketing team wants to know the worldwide percentage of consumers who intend to purchase the new model, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage?

a. Assume that nothing is known about the worldwide percentage of consumers who intend to buy the new model.

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Textbook Question

Large Data Sets from Appendix B. In Exercises 21 and 22, use the data set in Appendix B. Assume that each sample is a simple random sample obtained from a population with a normal distribution.

Birth Weights Refer to Data Set 6 “Births” in Appendix B.

a. Use the 205 birth weights of girls to construct a 95% confidence interval estimate of the standard deviation of the population from which the sample was obtained.

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Textbook Question

E-Cigarettes A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.7% of them being regular users of e-cigarettes. Because e-cigarette use is relatively new, there is a need to obtain today’s usage rate. How many adults must be surveyed now if we want a confidence level of 95% and a margin of error of 1.5 percentage points?

b. Use the results from the 2014 survey.

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Textbook Question

Voting Survey In a survey of 1002 people, 70% said that they voted in a recent presidential election (based on data from ICR Research Group). Voting records show that 61% of eligible voters actually did vote.

b. Find a 95% confidence interval estimate of the percentage of people who say that they voted.

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Textbook Question

No Failures According to the Rule of Three, when we have a sample size n with x=0 successes, we have 95% confidence that the true population proportion has an upper bound of 3/n. (See “A Look at the Rule of Three,” by Jovanovic and Levy, American Statistician, Vol. 51, No. 2.)

b. In a study of failure rates of computer hard drives, 45 Toshiba model MD04ABA500V hard drives were tested and there were no failures. What is the 95% upper bound for the percentage of failures for the population of all such hard drives?

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