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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 6.3.4
Chapter 6, Problem 6.3.4

"Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.
Social Security In a survey of 351 retired Americans, 200 said that they rely on Social Security as major source of income. (Adapted from Gallup)"

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1
Identify the given values: The total number of surveyed retired Americans is 351, and the number of those who rely on Social Security as a major source of income is 200.
Understand the formula for the point estimate of the population proportion (p^): p^ = x / n, where x is the number of successes (in this case, the number of people relying on Social Security) and n is the total sample size.
Substitute the given values into the formula: p^ = 200 / 351.
Calculate the complement of the population proportion (q^): q^ = 1 - p^. This represents the proportion of people who do not rely on Social Security as a major source of income.
Summarize the results: p^ is the point estimate for the proportion of people relying on Social Security, and q^ is the complement proportion. These values are derived from the sample data.

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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, denoted as p, represents the fraction of a population that possesses a certain characteristic. In this context, it refers to the proportion of retired Americans who rely on Social Security as a major source of income. Understanding this concept is crucial for estimating how widespread a particular opinion or behavior is within a larger group.
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Finding a Confidence Interval for a Population Proportion Using a TI84

Point Estimate

A point estimate is a single value that serves as an approximation of a population parameter. In this case, p^ (p-hat) is the point estimate of the population proportion p, calculated from sample data. It provides a quick and straightforward way to summarize the data, allowing for easier interpretation and communication of results.
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Introduction to Confidence Intervals

Complement of a Proportion

The complement of a proportion, denoted as q, represents the proportion of the population that does not have the characteristic of interest. In this scenario, q would be the proportion of retired Americans who do not rely on Social Security as a major source of income. Understanding both p and q is essential for a complete analysis of the survey results.
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Difference in Proportions: Confidence Intervals
Related Practice
Textbook Question

In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.

(0.512, 0.596)

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Textbook Question

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.

In a survey of 1502 U.S. adults, 31% said that they use Pinterest. The survey’s margin of error is ±2.9%. (Source: Pew Research Center)

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Textbook Question

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.

c = 0.95, s^2 = 11.56, n = 30

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Textbook Question

The data set represents the amounts of time (in minutes) spent checking email for a random sample of employees at a company.

c. Repeat part (b), assuming σ = 3.5 minutes. Compare the results.

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Textbook Question

[APPLET] The winning times (in hours) for a sample of 20 randomly selected Boston Marathon Women’s Open Division champions from 1980 to 2019 are shown in the table at the left. Assume the population standard deviation is 0.068 hour. (Source: Boston Athletic Association)

a. Find the point estimate of the population mean.

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Textbook Question

You wish to estimate the mean winning time for Boston Marathon Women’s Open Division champions. The estimate must be within 2 minutes of the population mean. Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Use the population standard deviation from Exercise 1.

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