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.8

Chapter 6, Problem 6.R.8

Determine the minimum sample size required to be 99% confident that the sample mean driving distance to work is within 2 miles of the population mean driving distance to work. Use the population standard deviation from Exercise 2.

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Identify the key components of the problem: the confidence level (99%), the margin of error (2 miles), and the population standard deviation (σ), which should be provided in Exercise 2. Denote the sample size as n.

Determine the z-score corresponding to a 99% confidence level. For a 99% confidence level, the z-score is the critical value that leaves 0.5% in each tail of the standard normal distribution. Use a z-table or statistical software to find this value.

Use the formula for the margin of error in estimating a population mean: \( E = z \cdot \frac{\sigma}{\sqrt{n}} \), where E is the margin of error, z is the z-score, σ is the population standard deviation, and n is the sample size.

Rearrange the formula to solve for the sample size n: \( n = \left( \frac{z \cdot \sigma}{E} \right)^2 \). Substitute the values for z, σ, and E (2 miles) into the formula.

Perform the calculations to determine 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 the process of calculating the number of observations or replicates needed in a statistical study to ensure that the results are reliable and valid. It is influenced by the desired confidence level, the margin of error, and the population standard deviation. A larger sample size generally leads to more accurate estimates of the population parameters.

Confidence Level

The confidence level represents the degree of certainty that the population parameter lies within a specified range of the sample statistic. A 99% confidence level indicates that if the same sampling procedure were repeated multiple times, approximately 99% of the calculated confidence intervals would contain the true population mean. This high level of confidence typically requires a larger sample size.

Margin of Error

The margin of error is the range within which the true population parameter is expected to lie, based on the sample statistic. In this context, a margin of error of 2 miles means that the sample mean driving distance should be within 2 miles of the actual population mean. The margin of error is influenced by the sample size and the variability of the data, with smaller margins requiring larger samples.

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Finding the Minimum Sample Size Needed for a Confidence Interval

Related Practice

Textbook Question

(a) Construct a 90% confidence interval for the population mean in Exercise 1. Interpret the results. (b) Does it seem likely that the population mean could be within 10% of the sample mean? Explain.

Textbook Question

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.98, n = 25

Textbook Question

In a random sample of 36 top-rated roller coasters, the average height is 165 feet and the standard deviation is 67 feet. Construct a 90% confidence interval for μ. Interpret the results. (Source: POP World Media, LLC)

Textbook Question

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

(7.428, 7.562)

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Textbook Question

In Exercises 19–22, let p be the population proportion for the situation. (a) Find point estimates of p and q, (b) construct 90% and 95% confidence intervals for p, and (c) interpret the results of part (b) and compare the widths of the confidence intervals.

In a survey of 912 U.S. adults in Generation Z (born after 1996), 383 said they are at least somewhat likely to consider an electric vehicle for their next vehicle purchase. (Adapted from Pew Research Center)

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.

a. No preliminary estimate is available. Find the minimum sample size needed.