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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.1b
Chapter 6, Problem 6.T.1b

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)
b. Construct a 95% confidence interval for the population proportion. Interpret the results.

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Step 1: Identify the sample proportion (p̂). The sample proportion is calculated as the number of successes (individuals who think players should take time off) divided by the total sample size. Use the formula: p̂ = x / n, where x = 1740 and n = 2096.
Step 2: Calculate the standard error (SE) for the sample proportion. The formula for the standard error is: SE = sqrt((p̂ * (1 - p̂)) / n). Substitute the value of p̂ from Step 1 and the sample size n = 2096 into this formula.
Step 3: Determine the critical value (z*) for a 95% confidence level. For a 95% confidence interval, the critical value z* is approximately 1.96 (based on the standard normal distribution).
Step 4: Compute the margin of error (ME). The margin of error is calculated using the formula: ME = z* * SE. Use the z* value from Step 3 and the SE from Step 2 to find the margin of error.
Step 5: Construct the confidence interval. The confidence interval is given by: p̂ ± ME. Add and subtract the margin of error from the sample proportion to find the lower and upper bounds of the interval. Finally, interpret the results by explaining that you are 95% confident the true population proportion lies within this interval.

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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 exhibits a certain characteristic. In this case, it represents the proportion of U.S. adults who believe that football players should take time off after a head injury. Understanding this concept is crucial for estimating how widespread this belief is among the entire population based on survey data.
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Constructing Confidence Intervals for Proportions

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, such as 95%. It provides an estimate of uncertainty around the sample proportion and helps in making inferences about the population. Constructing a confidence interval involves using the sample proportion and the standard error to determine the range.
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Introduction to Confidence Intervals

Standard Error

The standard error measures the variability of a sample statistic, such as the sample proportion, from the true population parameter. It is calculated using the sample size and the sample proportion, and it plays a critical role in constructing confidence intervals. A smaller standard error indicates that the sample proportion is a more accurate estimate of the population proportion.
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Calculating Standard Deviation
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.

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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.

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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)

d. Determine the minimum sample size required to be 95% confident that the sample mean test score is within 10 points of the population mean test score.

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Textbook Question

Use the standard normal distribution or the t-distribution to construct the indicated confidence interval for the population mean of each data set. Justify your decision. If neither distribution can be used, explain why. Interpret the results.

b. In a random sample of 15 cereal boxes, the mean weight was 11.89 ounces. Assume the weights of the cereal boxes are normally distributed and the population standard deviation is 0.05 ounce. Construct a 90% confidence interval for the population mean.

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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.

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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

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.

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