Textbook Question
In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.
(0.512, 0.596)
Ch. 6 - Confidence Intervals
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
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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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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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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08:09Difference in Proportions: Confidence Intervals
Related Practice
Textbook Question
True or False? In Exercises 1 and 2, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
The point estimate for the population proportion of failures is 1-p^
Textbook Question
In Exercises 13–16, find the margin of error for the values of c, σ and n.
c = 0.975, σ = 4.6, n = 100
Textbook Question
Bisexual Idenfitication In a survey of 692 lesbian, gay, bisexual, or transgender U.S adults, 378 said that they consider themselves bisexual. Construct a 90% confidence interval for the population proportion of lesbian, gay, bisexual, or transgender U.S. adults who consider themselves bisexual. (Adapted from Gallup)
Textbook Question
Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.
c = 0.98
Textbook Question
Finding the Margin of Error In Exercises 33 and 34, use the confidence interval to find the estimated margin of error. Then find the sample mean. Book Prices A store manager reports a confidence interval of (244.07, 280.97) when estimating the mean price (in dollars) for the population of textbooks.