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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1:58 minutes

Problem 6.3.1

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^

Verified step by step guidance

Understand the problem: The statement is about the point estimate for the population proportion of failures. Recall that the population proportion of successes is denoted by p, and the population proportion of failures is complementary to this, represented as 1 - p.

Review the concept: The point estimate for the population proportion of failures is derived from the complement of the population proportion of successes. If p represents the proportion of successes, then 1 - p represents the proportion of failures.

Analyze the statement: The statement claims that the point estimate for the population proportion of failures is 1 - p^ (where p^ is the sample proportion of successes). This aligns with the definition of the complement rule in probability.

Determine the truth value: Since the statement correctly describes the relationship between the sample proportion of successes (p^) and the sample proportion of failures (1 - p^), the statement is true.

Conclude: The statement is true, and no rewriting is necessary. If the statement were false, you would need to correct it by ensuring the complement relationship is properly stated as 1 - p^ for the sample proportion of failures.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Point Estimate

A point estimate is a single value that serves as an approximation of a population parameter. In statistics, it is often derived from sample data and is used to infer characteristics about the entire population. For example, the sample mean is a point estimate of the population mean.

Population Proportion

The population proportion refers to the fraction of a population that possesses a certain characteristic. It is denoted by 'p' and is crucial in understanding the distribution of categorical data. For instance, if 30 out of 100 surveyed individuals prefer a certain product, the population proportion of preference is 0.3.

Complement of a Proportion

The complement of a proportion is the probability that an event does not occur, calculated as 1 minus the proportion of the event. In the context of failures, if 'p' represents the proportion of successes, then '1 - p' represents the proportion of failures. This concept is essential for accurately interpreting and calculating probabilities in statistical analysis.

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Textbook Question

In Exercise 19, would it be unusual for the population proportion to be 38%? Explain.

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Textbook Question

Alcohol-Impaired Driving You wish to estimate, with 95% confidence, the population proportion of motor vehicle fatalities that were caused by alcohol-impaired driving. Your estimate must be accurate within 5% of the population proportion.

a. No preliminary estimate is available. Find the minimum sample size needed

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Textbook Question

b. Find the minimum sample size needed, using a prior study that found that 28% of motor vehicle fatalities were caused by alcohol-impaired driving. (Source: National Highway Traffic Safety Administration)

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Textbook Question

In Exercise 11, would it be unusual for the population proportion to be 72.5%? Explain.

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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 661 non-retired Americans, 218 said that they expect to rely on Social Security as major source of income when they retire. (Adapted from Gallup)

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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.

Private Internet Browsing In a survey of 4272 U.S. adults, 1025 knew that private browsing mode only prevents someone using the same computer from seeing one’s online activities. (Adapted from Pew Research Center)

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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

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

(0.087, 0.263)

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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^

Key Concepts

Point Estimate

Population Proportion

Complement of a Proportion

Watch next

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^