Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.19b

Chapter 6, Problem 6.3.19b

Fast Food You wish to estimate, with 90% confidence, the population proportion of U.S. families who eat fast food at least once per week. Your estimate must be accurate within 3% of the population proportion.
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)

Verified step by step guidance

Step 1: Identify the formula for calculating 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. From the problem, the confidence level is 90%, so the z-score (Z) corresponding to 90% confidence is approximately 1.645. The estimated population proportion (p) is 0.83, and the margin of error (E) is 0.03.

Step 3: Substitute the values into the formula. Replace Z with 1.645, p with 0.83, and E with 0.03 in the formula: n = (1.645^2 * 0.83 * (1 - 0.83)) / 0.03^2.

Step 4: Simplify the expression step by step. First, calculate Z^2 (1.645^2), then calculate p * (1 - p) (0.83 * (1 - 0.83)), and finally divide the product of these values by E^2 (0.03^2).

Step 5: Round the result up to the nearest whole number, as the sample size must be a whole number. This will give you the minimum sample size needed to estimate the population proportion with the specified confidence level and margin of error.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Population Proportion

The population proportion refers to the fraction of a population that possesses a certain characteristic. In this context, it is the percentage of U.S. families that eat fast food at least once per week. Understanding this concept is crucial for estimating the true proportion based on sample data.

Sample Size Calculation

Sample size calculation is the process of determining the number of observations or replicates needed to ensure that the sample accurately reflects the population. It involves factors such as the desired confidence level, margin of error, and the estimated population proportion, which in this case is 83%.

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. In this scenario, a 90% confidence level indicates that if the study were repeated multiple times, 90% of the calculated intervals would contain the true population proportion.

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Introduction to Confidence Intervals

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 a Confidence Interval In Exercises 25–28, use the data set to (a) find the sample mean. Assume the population is normally distributed.

Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students

Textbook Question

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)

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.

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.

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

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)