Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.17b

Chapter 6, Problem 6.3.17b

Congress You wish to estimate, with 95% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Your estimate must be accurate within 4% of the population proportion.
b. Find the minimum sample size needed, using a prior survey that found that 21% of likely U.S. voters think Congress is doing a good or excellent job. (Source: Rasmussen Reports)

Verified step by step guidance

Step 1: Identify the formula for determining the minimum sample size for estimating a population proportion. The formula is: n = (Z^2 * p * (1 - p)) / E^2, where n is the sample size, Z is the z-score corresponding to the confidence level, p is the estimated population proportion, and E is the margin of error.

Step 2: Determine the values for the variables in the formula. For a 95% confidence level, the z-score (Z) is approximately 1.96. The estimated population proportion (p) is 0.21 (from the prior survey), and the margin of error (E) is 0.04 (4%).

Step 3: Substitute the values into the formula. This gives: n = (1.96^2 * 0.21 * (1 - 0.21)) / 0.04^2.

Step 4: Simplify the numerator of the formula. Calculate 1.96^2, then multiply it by 0.21 and (1 - 0.21), which is 0.79.

Step 5: Simplify the denominator of the formula. Calculate 0.04^2, then divide the simplified numerator by the simplified denominator to find the minimum sample size. 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.

Population Proportion

Population proportion refers to the fraction of a population that exhibits a certain characteristic. In this context, it is the percentage of likely U.S. voters who believe Congress is performing well. Understanding this concept is crucial for estimating how representative a sample will be of the entire population.

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed to ensure that the results of a survey or experiment are statistically valid. It takes into account the desired confidence level, margin of error, and the estimated population proportion, which in this case is 21%.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. In this scenario, a 95% confidence level means that if the survey were repeated multiple times, 95% of the calculated intervals would contain the true population proportion of voters' opinions on Congress.

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Related Practice

Textbook Question

When all other quantities remain the same, how does the indicated change affect the minimum sample size requirement? Explain.

b. Increase in the error tolerance

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Textbook Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (b) the population standard deviation σ. Interpret the results.

[APPLET] Earnings The annual earnings (in thousands of dollars) of 21 randomly selected level 1 computer hardware engineers are listed. Use a 99% level of confidence. (Adapted from Salary.com)

100

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Textbook Question

Car Batteries The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence.

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Textbook Question

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Textbook Question

Alcohol-Impaired Driving You wish to estimate, with 95% confidence, the population proportion of motor vehicle fatalities that were caused by alcohol-impaired driving. 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 28% of motor vehicle fatalities were caused by alcohol-impaired driving. (Source: National Highway Traffic Safety Administration)

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Textbook Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.

Volleyball The numbers of service aces scored by 15 teams randomly selected from the top 50 NCAA Division I Women’s Volleyball teams for the 2021 season have a sample standard deviation of 26.1. Use an 80% level of confidence. (Source: National Collegiate Athletic Association)

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