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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
All textbooksLarson 8th EditionCh. 8 - Hypothesis Testing with Two SamplesProblem 8.Q.1c
Chapter 8, Problem 8.Q.1c

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).

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
1
Identify the type of hypothesis test: Since we are comparing the means of two independent samples with known population standard deviations, this is a two-sample z-test for the difference between means.
Set up the null and alternative hypotheses: The null hypothesis \(H_0\) is that the mean score for males is equal to or greater than the mean score for females, i.e., \(\mu_{male} \geq \mu_{female}\). The alternative hypothesis \(H_a\) is that the mean score for males is less than the mean score for females, i.e., \(\mu_{male} < \mu_{female}\). This indicates a left-tailed test.
Determine the significance level \(\alpha\): Given \(\alpha = 0.05\), and since this is a left-tailed test, the critical value corresponds to the z-score where the cumulative probability is 0.05.
Find the critical value: Use the standard normal distribution table or a calculator to find the z-score \(z_{\alpha}\) such that \(P(Z < z_{\alpha}) = 0.05\). This critical value marks the boundary of the rejection region.
Identify the rejection region: For a left-tailed test at \(\alpha = 0.05\), the rejection region consists of all z-scores less than the critical value \(z_{\alpha}\). If the calculated test statistic falls into this region, 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.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to support a specific claim about a population parameter. It involves formulating a null hypothesis (no effect or difference) and an alternative hypothesis (the claim to test), then using sample data to determine whether to reject the null hypothesis at a given significance level.
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Performing Hypothesis Tests: Proportions

Critical Value and Rejection Region

The critical value is a threshold that defines the boundary of the rejection region in hypothesis testing. It depends on the significance level (α) and the type of test (one-tailed or two-tailed). If the test statistic falls into the rejection region beyond the critical value, the null hypothesis is rejected in favor of the alternative.
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Critical Values: t-Distribution

Two-Sample Z-Test for Means with Known Population Standard Deviations

This test compares the means of two independent samples when population standard deviations are known. It calculates a Z statistic based on the difference between sample means, population standard deviations, and sample sizes. The test determines if the observed difference is statistically significant under the null hypothesis.
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Population Standard Deviation Known
Related Practice
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.

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

In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.


σ=0.63

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


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. (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.


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)