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

9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Proportions

Statistics

9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Proportions

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1:51 minutes

Problem 7.4.2

Textbook Question

Explain how to test a population proportion p.

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis typically states that the population proportion p is equal to a specific value (e.g., H₀: p = p₀), while the alternative hypothesis specifies whether p is different, greater, or less than p₀ (e.g., Hₐ: p ≠ p₀, Hₐ: p > p₀, or Hₐ: p < p₀).

Step 2: Choose the significance level (α), which represents the probability of rejecting the null hypothesis when it is true. Common values for α are 0.05, 0.01, or 0.10.

Step 3: Calculate the test statistic using the formula:

(p - p

₀)p₀(1-p₀)n, where p is the sample proportion, p₀ is the hypothesized population proportion, and n is the sample size.

Step 4: Determine the critical value(s) or p-value based on the chosen significance level and the type of test (one-tailed or two-tailed). Use the standard normal distribution (Z-distribution) to find these values.

Step 5: Compare the test statistic to the critical value(s) or use the p-value to make a decision. If the test statistic falls in the rejection region or if the p-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

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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 in a population that possess a certain characteristic. It is a key parameter in statistics, particularly in hypothesis testing and confidence interval estimation. Understanding the population proportion is essential for determining how representative a sample is of the entire population.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In the context of testing a population proportion, it involves formulating a null hypothesis (H0) that states there is no effect or difference, and an alternative hypothesis (H1) that suggests otherwise. The outcome of the test helps determine whether to reject or fail to reject the null hypothesis.

Z-Test for Proportions

The Z-test for proportions is a statistical test used to determine if the observed sample proportion significantly differs from a hypothesized population proportion. This test calculates a Z-score, which measures how many standard deviations the sample proportion is from the hypothesized proportion. It is essential for assessing the significance of the results and making informed conclusions about the population.

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Related Practice

Multiple Choice

Perform a 2-tailed hypothesis test for the true proportion of successes using the given values:

$alpha equals 0.10$ , $n equals 100$ , $x equals 42$ , & claim is $p equals 0.25$

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Multiple Choice

A local park claims that less than 15% of visitors litter. A random sample of 120 visitors finds that 25 litter. At the 0.05 significance level, test if the proportion of visitors who litter is greater than 15%.

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Textbook Question

Feeling Your Age A research organization conducts a survey by randomly selecting adults and asking each, “How do you feel relative to your age?” The results are shown in the figure. (Adapted from Pew Research Center)

[IMAGE]

a. Use a sign test to test the null hypothesis that the proportion of adults who feel older is equal to the proportion of adults who feel younger. Assign a + sign to each adult who responded “older,” assign a - sign to each adult who responded “younger,” and assign a 0 to each adult who responded “my age.” Use α = 0.05

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Textbook Question

Contacting Parents A research organization conducts a survey by randomly selecting adults and asking each, “How frequently do you contact your parents by phone?” The results are shown in the figure. (Adapted from Pew Research Center)

a. Use a sign test to test the null hypothesis that the proportion of adults who contact their parents by phone weekly is equal to the proportion of adults who contact their parents by phone daily. Assign a + sign to each adult who responded “weekly,” assign a - sign to each adult who responded “daily,” and assign a 0 to each adult who responded “other.” Use α = 0.05

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Textbook Question

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p <0.12, α=0.01. Sample statistics: p_hat = 0.10, n=40

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Textbook Question

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p ≥0.48, α=0.08. Sample statistics: p_hat = 0.40, n=90

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Textbook Question

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.

Claim: p ≠0.15, α=0.05. Sample statistics: p_hat = 0.12, n=500