Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problem 8.Q.1b

Chapter 8, Problem 8.Q.1b

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)

Verified step by step guidance

Identify the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). Since the claim is that the mean score for male students is less than that for female students, set up the hypotheses as: \(H_0: \mu_{male} = \mu_{female}\) and \(H_a: \mu_{male} < \mu_{female}\).

Determine the type of test based on the hypotheses. Because the alternative hypothesis is looking for a difference in one direction (male mean less than female mean), this is a left-tailed test.

Decide whether to use a z-test or a t-test. Since the population standard deviations are known (41 for males and 39 for females) and the sample sizes are both large (49 and 50), use a z-test.

Summarize: This is a left-tailed z-test comparing two independent sample means with known population standard deviations.

Next steps (not requested here) would involve calculating the test statistic using the formula for the difference between two means with known population standard deviations: \(Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\), where \(\bar{x}_1\) and \(\bar{x}_2\) are sample means, \(\sigma_1\) and \(\sigma_2\) are population standard deviations, and \(n_1\), \(n_2\) are sample sizes.

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Key Concepts

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

Hypothesis Testing and Tail Types

Hypothesis testing involves making a claim about a population parameter and using sample data to assess its validity. The test can be left-tailed, right-tailed, or two-tailed depending on the alternative hypothesis. A left-tailed test checks if a parameter is less than a value, a right-tailed if greater, and two-tailed if different (either less or greater). In this question, since the claim is that male scores are less than female scores, a left-tailed test is appropriate.

Z-test vs. T-test

A z-test is used when the population standard deviation is known and the sample size is large (usually n > 30), while a t-test is used when the population standard deviation is unknown and/or the sample size is small. Here, population standard deviations are given and sample sizes are 49 and 50, so a z-test is appropriate. Choosing the correct test ensures valid conclusions about the population means.

Comparing Two Independent Means

When comparing means from two independent groups, the difference between sample means is analyzed to infer about population means. The test statistic considers the means, standard deviations, and sample sizes of both groups. This approach helps determine if observed differences are statistically significant or due to random chance, which is essential for supporting or rejecting the claim about male and female reading scores.

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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.

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.

c. Find the critical value(s) and identify the rejection region(s).

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

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.

c. Find the critical value(s) and identify the rejection region(s).

d. Find the appropriate standardized test statistic.

e. Decide whether to reject or fail to reject the null hypothesis.

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

[APPLET] The table shows the credit scores for 12 randomly selected adults who are considered high-risk borrowers before and two years after they attend a personal finance seminar. At α=0.01, is there enough evidence to support the claim that the personal finance seminar helps adults increase their credit scores? Assume the populations are normally distributed.

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

Textbook Question

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)

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

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.

c. Find the critical value(s) and identify the rejection region(s).

d. Find the appropriate standardized test statistic.

e. Decide whether to reject or fail to reject the null hypothesis.

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

A music teacher claims that the mean scores on a music assessment test for eighth grade students in public and private schools are equal. The mean score for 13 randomly selected public school students is 146 with a standard deviation of 49, and the mean score for 15 randomly selected private school students is 160 with a standard deviation of 42. At α=0.1, can you reject the teacher’s claim? Assume the populations are normally distributed and the population variances are equal. (Adapted from National Center for Education Statistics)