Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

Statistics

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

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2:17 minutes

Problem 6.3.3

Textbook Question

"Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.
Tax Fraud In a survey of 1040 U.S. adults, 62 have had someone impersonate them to try to claim tax refunds. (Adapted from Pew Research Center)"

Verified step by step guidance

Identify the given information: The total number of U.S. adults surveyed is 1040, and the number of adults who have had someone impersonate them to claim tax refunds is 62.

Understand the concept: The point estimate for the population proportion (p̂) is calculated as the ratio of the number of successes (in this case, adults impersonated) to the total sample size. The formula is p̂ = x / n, where x is the number of successes and n is the total sample size.

Substitute the given values into the formula for p̂: Use x = 62 and n = 1040 to calculate p̂.

Find q̂, the complement of p̂: Recall that q̂ = 1 - p̂. Once p̂ is calculated, subtract it from 1 to find q̂.

Interpret the results: p̂ represents the proportion of adults in the sample who have been impersonated for tax fraud, and q̂ represents the proportion of adults in the sample who have not been impersonated for tax fraud.

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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 U.S. adults who have experienced impersonation for tax fraud. Understanding this concept is crucial for estimating how widespread the issue is within the entire population based on survey data.

Point Estimate

A point estimate is a single value derived from sample data that serves as a best guess for a population parameter. In this case, p^ (p-hat) is the point estimate of the population proportion p, calculated by dividing the number of individuals with the characteristic (62) by the total sample size (1040). This estimate provides a quick snapshot of the population's behavior regarding tax fraud.

Complement of the Population Proportion

The complement of the population proportion, denoted as q, represents the proportion of the population that does not have the characteristic in question. It is calculated as q = 1 - p. In the context of the survey, q would indicate the proportion of U.S. adults who have not experienced impersonation for tax fraud, providing a complete view of the situation.

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Related Practice

Textbook Question

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)

d. Find the minimum sample size needed to estimate the population proportion at the 99% confidence level to ensure that the estimate is accurate within 3% of the population proportion.

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Multiple Choice

Make a confidence interval for $p$ given the following values.

$x = 314, n = 500, C = 99 %$

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Multiple Choice

Make a confidence interval for $p$ given the following values.

$\hat{p} = 0.15, n = 60, C = 90 %$

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Multiple Choice

A health clinic is studying the effectiveness of a new flu vaccine. In a random sample of 80 patients who received the vaccine, 68 did not get the flu during the season. Make a 99% confidence interval for the true proportion of vaccinated individuals who are protected from the flu.

We are ___%% confident that between (_––––,_––––) of vaccinated individuals are protected from the flu.

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Textbook Question

"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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Multiple Choice

Over the first $20$ days of the semester, one student is late to class on $6$ days. Find the margin of error for a $98 percent sign$ confidence interval for the true proportion of time this student is late.

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Multiple Choice

Over the first $20$ days of the semester, one student is late to class on $6$ days. Construct a $98 percent sign$ confidence interval for the true proportion of time this student is late.

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Multiple Choice

A previous study found that your school consists of $60\%$ White/Caucasian students. You want the $98\%$ confidence interval for the proportion of White/Caucasian students to be no more than $.05$ away from the true proportion. How many students must you include in a sample to create this confidence interval?