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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.C.13
Chapter 8, Problem 8.C.13

An education organization claims that the mean SAT scores for male athletes and male non-athletes at a college are different. A random sample of 26 male athletes at the college has a mean SAT score of 1189 and a standard deviation of 218. A random sample of 18 male non-athletes at the college has a mean SAT score of 1376 and a standard deviation of 186. At α=0.05, can you support the organization’s claim? Interpret the decision in the context of the original claim. Assume the populations are normally distributed and the population variances are equal.

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Identify the hypotheses for the test. The null hypothesis \(H_0\) states that the mean SAT scores for male athletes and male non-athletes are equal: \(\mu_1 = \mu_2\). The alternative hypothesis \(H_a\) states that the means are different: \(\mu_1 \neq \mu_2\).
Since the population variances are assumed equal and the samples are independent, use the two-sample t-test for equal variances. Calculate the pooled standard deviation \(s_p\) using the formula: \[s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}\] where \(n_1\) and \(n_2\) are the sample sizes, and \(s_1\) and \(s_2\) are the sample standard deviations.
Calculate the test statistic \(t\) using the formula: \[t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}\] where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means.
Determine the degrees of freedom for the test, which is \(df = n_1 + n_2 - 2\). Then, find the critical t-value(s) from the t-distribution table for a two-tailed test at significance level \(\alpha = 0.05\).
Compare the absolute value of the calculated test statistic \(|t|\) to the critical t-value. If \(|t|\) is greater than the critical value, reject the null hypothesis \(H_0\) and conclude there is sufficient evidence to support the claim that the mean SAT scores differ. Otherwise, do not reject \(H_0\). Interpret this decision in the context of the original claim about the difference in mean SAT scores.

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Key Concepts

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

Two-Sample t-Test for Means

This test compares the means of two independent groups to determine if there is a statistically significant difference between them. It uses sample means, standard deviations, and sizes to calculate a t-statistic, which is then compared to a critical value based on the chosen significance level (α). Here, it helps test if male athletes and non-athletes have different mean SAT scores.
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Sampling Distribution of Sample Mean

Assumption of Equal Population Variances

When performing a two-sample t-test, assuming equal variances means the variability in both groups is similar. This allows pooling the sample variances to get a more accurate estimate of the common variance, which affects the calculation of the test statistic and degrees of freedom. This assumption simplifies the test and is crucial for valid results.
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Population Standard Deviation Known

Significance Level and Hypothesis Testing

The significance level (α = 0.05) defines the threshold for rejecting the null hypothesis, which states there is no difference between group means. If the test statistic falls in the critical region, we reject the null and support the claim of a difference. Interpreting the decision involves relating statistical results back to the real-world context of SAT scores.
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Performing Hypothesis Tests: Proportions
Related Practice
Textbook Question

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?


c=0.90, x̅=8.21, σ=0.62, n=8

Textbook Question

Testing the Difference Between Two Means In Exercises 15–24, (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.

ACT Mathematics and Science Scores The mean ACT mathematics score for 60 high school students is 20.2. Assume the population standard deviation is 5.7. The mean ACT science score for 75 high school students is 20.6. Assume the population standard deviation is 5.9. At α=0.01, can you reject the claim that ACT mathematics and science scores are equal? (Source: ACT, Inc.)

Textbook Question

Find the critical value(s) for the alternative hypothesis, level of significance , and sample sizes and . Assume that the samples are random and independent, the populations are normally distributed, and the population variances are (a) equal and (b) not equal.

Ha:μ1>μ2 , α=0.01 , n1=12 , n2=15

Textbook Question

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?


c=0.95, x̅=3.46, s=1.63, n=16

Textbook Question

In a survey of 4860 U.S. adults, 77% said they would date or have already dated someone whose religion was different from theirs. (Source: Pew Research Center)


Construct a 95% confidence interval for the proportion of U.S. adults who say they would date or have already dated someone whose religion was different from theirs.

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



A researcher claims that the mean annual earnings for locksmiths is \$55,000. At α=0.05, can you reject the researcher’s claim? Interpret the decision in the context of the original claim.