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

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.

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Textbook Question

Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (b) find the sample standard deviation

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Textbook Question

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b. Find the minimum sample size needed, using a prior study that found that 83% of U.S. families eat fast food at least once per week. (Source: The Barbecue Lab)

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)

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)