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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.2
Chapter 8, Problem 8.Q.2

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)

Verified step by step guidance
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Step 1: Identify the claim and state the null hypothesis (Ho) and the alternative hypothesis (Ha). The claim is that the mean scores on the music assessment test for eighth grade students in public and private schools are equal. Therefore, Ho: μ1 = μ2 (the means are equal) and Ha: μ1 ≠ μ2 (the means are not equal).
Step 2: Determine the type of test and tail direction. Since the alternative hypothesis (Ha) is μ1 ≠ μ2, this is a two-tailed test. Additionally, because the population variances are assumed to be equal and the sample sizes are small, a t-test should be used.
Step 3: Find the critical value(s) and identify the rejection region(s). Using the significance level α = 0.1 and degrees of freedom calculated as df = (n1 + n2 - 2), where n1 = 13 and n2 = 15, look up the critical t-value in the t-distribution table for a two-tailed test. The rejection regions will be in the tails beyond the critical values.
Step 4: Calculate the standardized test statistic. Use the formula for the t-test for two independent samples: t=(x1-x2)(sp²(1n1+1n2)), where sp is the pooled standard deviation calculated as (s1²(n1-1)+s2²(n2-1))n1+n2-2.
Step 5: Compare the calculated test statistic to the critical value(s) and make a decision. If the test statistic falls within the rejection region, reject the null hypothesis (Ho). Otherwise, fail to reject Ho. Finally, interpret the decision in the context of the original claim: if Ho is rejected, the teacher's claim that the mean scores are equal is not supported; if Ho is not rejected, there is insufficient evidence to refute the teacher's claim.

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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 population parameters based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents no effect or no difference, and the alternative hypothesis (Ha), which indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence in the sample data to reject H0 in favor of Ha.
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Step 1: Write Hypotheses

Types of Tests (Z-test vs. T-test)

The choice between a z-test and a t-test depends on the sample size and whether the population standard deviation is known. A z-test is typically used when the sample size is large (n > 30) or when the population standard deviation is known, while a t-test is used for smaller samples (n ≤ 30) or when the population standard deviation is unknown. Both tests assess the significance of the difference between sample means.
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Critical Values and Rejection Regions

Critical values are the threshold points that define the boundaries of the rejection region in hypothesis testing. Depending on the significance level (α) and the type of test (one-tailed or two-tailed), critical values help determine whether the test statistic falls into the rejection region. If the test statistic exceeds the critical value, the null hypothesis is rejected, indicating that the sample provides sufficient evidence against H0.
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Critical Values: t-Distribution
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.


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)

Textbook Question

[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)


48,69446,85642,91261,67271,11254,861


69,45471,84159,75169,61254,28452,166


66,36048,16465,27235,25061,12765,397


58,92558,91659,01753,07045,19969,941


69,49257,08553,82952,69268,29853,792



Construct a 95% confidence interval for the population mean annual earnings for locksmiths.

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

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)


Construct a 99% confidence interval for the population variance.