Textbook Question
In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.
(0.512, 0.596)
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Ch. 6 - Confidence Intervals
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 6, Problem 6.Q.2
You wish to estimate the mean winning time for Boston Marathon Women’s Open Division champions. The estimate must be within 2 minutes of the population mean. Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Use the population standard deviation from Exercise 1.
Verified step by step guidance1
Identify the formula for determining the minimum sample size for estimating a population mean: \( n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \), where \( Z \) is the critical value for the confidence level, \( \sigma \) is the population standard deviation, and \( E \) is the margin of error.
Determine the critical value \( Z \) for a 99% confidence level. For a 99% confidence interval, the critical value corresponds to the z-score that leaves 0.5% in each tail of the standard normal distribution. This value is approximately \( Z = 2.576 \).
Substitute the given margin of error \( E = 2 \) minutes into the formula. This represents the maximum allowable difference between the sample mean and the population mean.
Use the population standard deviation \( \sigma \) provided in Exercise 1. Substitute this value into the formula. If \( \sigma \) is not explicitly given in this problem, refer to Exercise 1 for its value.
Plug all the values (\( Z = 2.576 \), \( \sigma \), and \( E = 2 \)) into the formula \( n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \) and compute the result. Round up to the nearest whole number, as the 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.
Confidence Interval
A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For example, a 99% confidence interval suggests that if we were to take many samples and construct intervals, approximately 99% of those intervals would contain the true population mean.
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Introduction to Confidence Intervals
Sample Size Determination
Sample size determination involves calculating the number of observations needed in a sample to achieve a desired level of precision for estimates. In this context, it requires using the population standard deviation, the desired margin of error (2 minutes), and the confidence level (99%) to ensure the estimate is statistically valid.
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Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of estimating the mean winning time, the population standard deviation provides insight into how much individual winning times vary from the mean, which is crucial for calculating the sample size needed for the confidence interval.
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Related Practice
Textbook Question
"Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.
Social Security In a survey of 351 retired Americans, 200 said that they rely on Social Security as major source of income. (Adapted from Gallup)"
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Textbook Question
Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.
In a survey of 1502 U.S. adults, 31% said that they use Pinterest. The survey’s margin of error is ±2.9%. (Source: Pew Research Center)
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Textbook Question
The data set represents the amounts of time (in minutes) spent checking email for a random sample of employees at a company.
c. Repeat part (b), assuming σ = 3.5 minutes. Compare the results.
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Textbook Question
[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)
a. Find the point estimate of the population mean.
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Textbook Question
[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)
d. Does it seem likely that the population mean could be greater than 2.52 hours? Explain.
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