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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.T.1a
Chapter 6, Problem 6.T.1a

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.

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Identify the formula for the point estimate of a population proportion, which is given by \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successes (individuals with the desired characteristic) and \( n \) is the total sample size.
From the problem, note that \( x = 1740 \) (the number of adults who think players should take time off) and \( n = 2096 \) (the total number of surveyed adults).
Substitute the values of \( x \) and \( n \) into the formula: \( \hat{p} = \frac{1740}{2096} \).
Simplify the fraction to calculate the proportion. This will give you the point estimate for the population proportion.
Interpret the result: The point estimate represents the proportion of U.S. adults who believe football players should take time off after a head injury, based on the survey data.

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Key Concepts

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

Point Estimate

A point estimate is a single value that serves as a best guess or approximation of a population parameter. In this context, it refers to the proportion of U.S. adults who believe that football players should take time off after a head injury. The point estimate is calculated by dividing the number of individuals who support the idea by the total number of surveyed individuals.
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Introduction to Confidence Intervals

Population Proportion

The population proportion is the fraction of a population that possesses a certain characteristic. It is denoted by 'p' and is crucial for understanding the overall sentiment of a group. In this case, it represents the proportion of all U.S. adults who think players should take time off after a head injury, which can be estimated using survey data.
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Constructing Confidence Intervals for Proportions

Sample Size

Sample size refers to the number of observations or data points collected in a survey or study. A larger sample size generally leads to more reliable estimates of population parameters, as it reduces the margin of error. In this question, the sample size of 2096 adults is significant for calculating the point estimate and assessing the reliability of the findings.
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Related Practice
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)

b. Construct a 90% confidence interval for the population mean. Interpret the results.

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

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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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.