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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
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 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
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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.
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Constructing Confidence Intervals for Proportions

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%.
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Sampling Distribution of Sample Proportion

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

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.

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

Senate Filibuster You wish to estimate, with 99% confidence, the population proportion of U.S. adults who disapprove of the U.S Senate’s use of the filibuster. Your estimate must be accurate within 2% of the population proportion.

b. Find the minimum sample size needed, using a prior survey that found that 34% of U.S. adults disapprove of the U.S Senate’s use of the filibuster. (Source: Monmouth University)

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Textbook Question

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

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Textbook Question

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.

b. Increase in the sample size

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Textbook Question

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Textbook Question

Constructing a Confidence Interval In Exercises 25–28, use the data set to (c) construct a 99% confidence interval for the population mean. Assume the population is normally distributed.

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

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