Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.R.7

Chapter 6, Problem 6.R.7

Determine the minimum sample size required to be 95% confident that the sample mean waking time is within 10 minutes of the population mean waking time. Use the population standard deviation from Exercise 1.

Verified step by step guidance

Identify the formula for determining the minimum sample size for estimating a population mean:

n = (z * σ / E)^2

, where

n

is the sample size,

z

is the z-score corresponding to the confidence level,

σ

is the population standard deviation, and

E

is the margin of error.

Determine the z-score for a 95% confidence level. For a 95% confidence level, the z-score corresponds to the critical value where the cumulative probability is 0.975 (since 95% confidence leaves 2.5% in each tail).

Substitute the given margin of error,

E = 10

minutes, into the formula. This represents the maximum allowable difference between the sample mean and the population mean.

Use the population standard deviation from Exercise 1, denoted as

σ

, and substitute it into the formula. Ensure the units are consistent (e.g., minutes).

Calculate the value of

n

by squaring the result of

(z * σ / E)

. 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 the process of calculating the number of observations or replicates needed in a statistical study to achieve a desired level of confidence and precision. In this context, it involves using the desired margin of error, confidence level, and population standard deviation to find the minimum sample size that ensures the sample mean is within a specified range of the population mean.

Confidence Level

The confidence level represents the probability that the confidence interval calculated from the sample data will contain the true population parameter. A 95% confidence level indicates that if the same sampling procedure were repeated multiple times, approximately 95% of the calculated intervals would capture the true population mean, providing a strong assurance of the reliability of the results.

Margin of Error

The margin of error is the range within which the true population parameter is expected to lie, given a certain level of confidence. In this scenario, a margin of error of 10 minutes means that the sample mean waking time should be within 10 minutes of the actual population mean. This concept is crucial for determining how precise the estimate needs to be and directly influences the required sample size.

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Finding the Minimum Sample Size Needed for a Confidence Interval

Related Practice

Textbook Question

In Exercises 9–12, find the critical value tc for the level of confidence c and sample size n.

c = 0.98, n = 15

Textbook Question

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.99, s = 16.5, n = 20, xbar = 25.2

Textbook Question

In Exercises 5 and 6, use the confidence interval to find the margin of error and the sample mean.

(20.75, 24.10)

Textbook Question

In a random sample of 12 senior-level civil engineers, the mean annual earnings were \$133,326 and the standard deviation was \$36,729. Assume the annual earnings are normally distributed and construct a 95% confidence interval for the population mean annual earnings for senior-level civil engineers. Interpret the results. (Adapted from Salary.com)

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.

Textbook Question

You wish to estimate, with 95% confidence, the population proportion of U.S. adults who have taken or planned to take a winter vacation in a recent year. Your estimate must be accurate within 5% of the population proportion.

b. Find the minimum sample size needed, using a prior study that found that 32% of U.S. adults have taken or planned to take a winter vacation in a recent year. (Source: Rasmussen Reports)