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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1:39 minutes

Problem 6.3.5

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)

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the point estimates of p (the sample proportion of individuals who expect to rely on Social Security as a major source of income) and q (the complement of p, representing those who do not expect to rely on Social Security).

Step 2: Recall the formula for the sample proportion p̂. It is given by:

p^= \frac{x}{n}

, where x is the number of individuals with the characteristic of interest (218 in this case), and n is the total sample size (661 in this case).

Step 3: Substitute the given values into the formula for p̂. This will give you the point estimate for p̂:

p^= \frac{218}{661}

Step 4: Recall that q̂ is the complement of p̂, which can be calculated using the formula:

q^= 1 - p^

. Use the value of p̂ from Step 3 to compute q̂.

Step 5: Interpret the results. The values of p̂ and q̂ represent the proportions of the sample that expect to rely on Social Security and do not expect to rely on Social Security, respectively.

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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, is the ratio of members of a population that have a particular characteristic to the total number of members in that population. In this context, it represents the proportion of non-retired Americans who expect to rely on Social Security as a major source of income when they retire.

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 summary of the sample's characteristics, allowing for easier interpretation of the data.

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 of interest. It is calculated as q = 1 - p. In this scenario, q^ (q-hat) would indicate the proportion of non-retired Americans who do not expect to rely on Social Security as a major source of income.

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