Skip to main content
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 6.3.19a
Chapter 6, Problem 6.3.19a

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

Verified step by step guidance
1
Step 1: Recall the formula for determining the minimum sample size for estimating a population proportion when no preliminary estimate is available: n = (Z^2 * 0.25) / E^2. Here, Z is the critical value corresponding to the desired confidence level, and E is the margin of error.
Step 2: Identify the given values. The confidence level is 90%, so the critical value Z can be found using a Z-table or standard normal distribution. For a 90% confidence level, Z ≈ 1.645. The margin of error E is given as 0.03 (3%).
Step 3: Substitute the values into the formula. Use Z = 1.645, E = 0.03, and the maximum variability for the population proportion (p = 0.5, q = 1 - p = 0.5, so p * q = 0.25). The formula becomes n = (1.645^2 * 0.25) / 0.03^2.
Step 4: Simplify the numerator by squaring the Z-value and multiplying it by 0.25. Then, simplify the denominator by squaring the margin of error (E).
Step 5: Divide the simplified numerator by the simplified denominator to calculate the minimum sample size. Always round up to the nearest whole number, as sample size must be an integer.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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 exhibits a certain characteristic, in this case, U.S. families who eat fast food at least once per week. It is denoted by 'p' and is crucial for estimating how widespread a behavior or opinion is within a defined group.
Recommended video:
05:45
Constructing Confidence Intervals for Proportions

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed to achieve a desired level of precision in estimates. In this scenario, it involves using the desired confidence level (90%) and margin of error (3%) to ensure that the sample accurately reflects the population proportion.
Recommended video:
05:11
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. For this question, a 90% confidence interval means that if the same sampling method were repeated multiple times, 90% of the intervals would contain the true population proportion.
Recommended video:
06:33
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 (a) the population variance σ^2. Interpret the results.

Drive-Thru Times The times (in seconds) spent by a random sample of 28 customers in the drive-thru of a fast-food restaurant have a sample standard deviation of 56.1. Use a 98% level of confidence.

Textbook Question

Since 1935, the Gallup Organization has conducted public opinion polls in the United States and around the world. The table shows the results of Gallup’s World Affairs Poll of 2021, in which 1021 U.S. adults were polled. The remaining percentages not shown in the results are adults who were not sure.

[IMAGE]

Find the minimum sample size needed to estimate, with 95% confidence, the population proportion of adults who feel that China’s economic power is a critical or an important economic threat to the United States. Your estimate must be accurate within 2% of the population proportion.

1
views
Textbook Question

Since 1935, the Gallup Organization has conducted public opinion polls in the United States and around the world. The table shows the results of Gallup’s World Affairs Poll of 2021, in which 1021 U.S. adults were polled. The remaining percentages not shown in the results are adults who were not sure.

b. What was the greatest value you obtained for p^?

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.

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

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.

Annual Precipitation The annual precipitation amounts (in inches) of a random sample of 61 years for Chicago, Illinois, have a sample standard deviation of 6.46. Use a 98% level of confidence. (Source: National Oceanic and Atmospheric Administration)

Textbook Question

Since 1935, the Gallup Organization has conducted public opinion polls in the United States and around the world. The table shows the results of Gallup’s World Affairs Poll of 2021, in which 1021 U.S. adults were polled. The remaining percentages not shown in the results are adults who were not sure.

Use technology to find a 95% confidence interval for the population proportion of adults who

a. view foreign trade as an economic opportunity.