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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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)"
Verified step by step guidance1
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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Video duration:
2mKey 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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02:35
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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06:33Introduction 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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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^
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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
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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)
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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
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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.
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