Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problem 8.R.29

Chapter 8, Problem 8.R.29

In Exercises 29 and 30, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent.

A medical research team conducted a study to test the effect of a drug used to treat a type of inflammation. In the study, 68 subjects took the drug and 68 subjects took a placebo. The results are shown below. At α=0.05, can you reject the claim that the proportion of subjects who had at least 24 weeks of accrued remission is the same for the two groups? (Source: The New England Journal of Medicine)

Verified step by step guidance

Step 1: Identify the claim and state the null hypothesis (Ho) and alternative hypothesis (Ha). The claim is that the proportion of subjects who had at least 24 weeks of accrued remission is the same for the drug and placebo groups. Ho: p1 = p2 (the proportions are equal). Ha: p1 ≠ p2 (the proportions are not equal).

Step 2: Calculate the sample proportions for each group. For the drug group, the proportion is p1 = 19/68. For the placebo group, the proportion is p2 = 2/68. These proportions will be used in subsequent calculations.

Step 3: Determine the critical value(s) and rejection region(s) for the test. Since α = 0.05 and the test is two-tailed, find the z-critical values corresponding to α/2 = 0.025 in each tail. Use a z-table or standard normal distribution to find these values.

Step 4: Compute the standardized test statistic z. Use the formula: z = (p1 - p2) / sqrt(p̂(1-p̂)(1/n1 + 1/n2)), where p̂ = (x1 + x2) / (n1 + n2) is the pooled proportion, x1 and x2 are the number of successes in each group, and n1 and n2 are the sample sizes.

Step 5: Compare the calculated z-value to the critical values and decide whether to reject or fail to reject the null hypothesis. If the z-value falls in the rejection region, reject Ho; otherwise, fail to reject Ho. Interpret the decision in the context of the original claim: determine whether there is sufficient evidence to conclude that the proportions are different.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which states there is no effect or difference, and the alternative hypothesis (Ha), which suggests there is an effect or difference. In this context, the claim being tested is whether the proportion of subjects with remission is the same for both the drug and placebo groups.

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which in this case is set at 0.05. The rejection region is the range of values for the test statistic that would lead to rejecting H0. Understanding these concepts is crucial for determining whether the observed data provides sufficient evidence against the null hypothesis.

Standardized Test Statistic (z)

The standardized test statistic, often denoted as z, measures how many standard deviations an element is from the mean. In hypothesis testing, it is calculated using sample data to compare the observed proportion to the expected proportion under the null hypothesis. This statistic helps in determining whether the observed difference between groups is statistically significant, guiding the decision to reject or fail to reject the null hypothesis.

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Step 2: Calculate Test Statistic

Related Practice

Textbook Question

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.

Sample 1: The retail prices of 20 motorcycles

Sample 2: The retail prices of 20 minivans

Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

b. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.

The mean score on a reading assessment test for 49 randomly selected male high school students was 279. Assume the population standard deviation is 41. The mean score on the same test for 50 randomly selected female high school students was 292. Assume the population standard deviation is 39. At α=0.05, can you support the claim that the mean score on the reading assessment test for male high school students is less than the mean score for female high school students? (Adapted from National Center for Education Statistics)

Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

f. Interpret the decision in the context of the original claim.

Textbook Question

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.

Sample 1: The heights of 37 children

Sample 2: The heights of the same 37 children after 1 year

Textbook Question

In Exercises 9 and 10, (a) identify the claim and state Ho and Ha , (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

A researcher claims that the mean sodium content of sandwiches at Restaurant A is less than the mean sodium content of sandwiches at Restaurant B. The mean sodium content of 22 randomly selected sandwiches at Restaurant A is 670 milligrams. Assume the population standard deviation is 20 milligrams. The mean sodium content of 28 randomly selected sandwiches at Restaurant B is 690 milligrams. Assume the population standard deviation is 30 milligrams. At α=0.05, is there enough evidence to support the claim?

Textbook Question

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

a. Identify the claim and state Ho and Ha