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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.Tech.2
Chapter 6, Problem 6.Tech.2

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.
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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.

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Step 1: Identify the key components of the problem. We are tasked with finding the minimum sample size needed to estimate a population proportion with a 95% confidence level and a margin of error of 2% (0.02). The formula for the minimum sample size is: n = z2⁢p(1-p)E2, where z is the z-score, p is the estimated population proportion, and E is the margin of error.
Step 2: Determine the z-score for a 95% confidence level. The z-score corresponds to the critical value of the standard normal distribution for a 95% confidence level. This value is approximately 1.96.
Step 3: Use the given data to estimate the population proportion p. If no specific proportion is provided, use p = 0.5 as it maximizes the required sample size, ensuring a conservative estimate.
Step 4: Substitute the values into the formula. Using z = 1.96, p = 0.5, and E = 0.02, the formula becomes: n = 1.962⁢0.5(1-0.5)0.022.
Step 5: Simplify the expression to calculate the minimum sample size. First, square the z-score and the margin of error. Then, multiply the proportion values and divide by the squared margin of error. Round up the result 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.

Sample Size Determination

Sample size determination is a statistical process used to calculate the number of observations or replicates needed in a study to ensure that the results are reliable and valid. In the context of estimating population proportions, the sample size must be large enough to achieve a desired level of confidence and margin of error. The formula typically involves the population proportion estimate, the z-score corresponding to the confidence level, and the margin of error.
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Confidence Level

The confidence level represents the degree of certainty that the population parameter lies within the confidence interval calculated from the sample data. A 95% confidence level means that if the same sampling method were repeated multiple times, approximately 95% of the calculated intervals would contain the true population proportion. This concept is crucial for understanding the reliability of the estimates derived from sample data.
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Margin of Error

Margin of error quantifies the uncertainty associated with a sample estimate. It indicates the range within which the true population parameter is expected to fall, given a certain confidence level. In this case, a margin of error of 2% means that the estimate of the population proportion can vary by plus or minus 2% from the sample proportion, which is essential for assessing the precision of the estimate.
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Related Practice
Textbook Question

The data set represents the weights (in pounds) of 10 randomly selected black bears from northeast Pennsylvania. Assume the weights are normally distributed. (Source: Pennsylvania Game Commission)

c. Construct a 99% confidence interval for the population standard deviation. Interpret the results.

Textbook Question

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.

Textbook Question

In a survey of 2096 U.S. adults, 1740 think football teams of all levels should require players who suffer a head injury to take a set amount of time off from playing to recover. (Adapted from The Harris Poll)

a. Find the point estimate for the population proportion.

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

The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

c. Would it be unusual for the population mean to be under 575? Explain.

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.