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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 8.2.14c
Chapter 8, Problem 8.2.14c

Testing the Difference Between Two Means, (c) find the standardized test statistic t, 
Assume the samples are random and independent, and the populations are normally distributed.
Transactions
 A magazine claims that the mean amount spent by a customer at Burger Stop is greater than the mean amount spent by a customer at Fry World. The results for samples of customer transactions for the two fast food restaurants are shown at the left. At , α=0.05 can you support the magazine’s claim? Assume the population variances are equal.
Table comparing sample means, standard deviations, and sizes for two groups: Dogs and Cats.

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Identify the sample statistics given: For Dogs (group 1), the sample mean \( \bar{x}_1 = 255 \), sample standard deviation \( s_1 = 30 \), and sample size \( n_1 = 16 \). For Cats (group 2), the sample mean \( \bar{x}_2 = 231 \), sample standard deviation \( s_2 = 28 \), and sample size \( n_2 = 18 \).
Since the population variances are assumed equal, calculate the pooled standard deviation \( s_p \) using the formula: \[ s_p = \sqrt{ \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} } \]
Calculate the standard error (SE) of the difference between the two means using the pooled standard deviation: \[ SE = s_p \times \sqrt{ \frac{1}{n_1} + \frac{1}{n_2} } \]
Compute the test statistic \( t \) for the difference between two means with equal variances: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} \]
Determine the degrees of freedom (df) for the test: \[ df = n_1 + n_2 - 2 \] Use this \( t \)-value and \( df \) to compare against the critical value from the \( t \)-distribution at \( \alpha = 0.05 \) for a one-tailed test to decide whether to support the magazine's claim.

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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 assumes the samples are random, independent, and populations are normally distributed. The test statistic follows a t-distribution when population variances are unknown but assumed equal.
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Sampling Distribution of Sample Mean

Pooled Variance and Equal Variance Assumption

When population variances are assumed equal, the sample variances are combined to calculate a pooled variance. This pooled variance provides a more accurate estimate of the common variance, which is used in the denominator of the t-test formula to standardize the difference between sample means.
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Variance & Standard Deviation of Discrete Random Variables

Significance Level (α) and Hypothesis Testing

The significance level α (here 0.05) defines the threshold for rejecting the null hypothesis. If the calculated t-statistic falls in the critical region beyond this threshold, the null hypothesis is rejected, supporting the claim that one mean is greater than the other with a specified confidence level.
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Performing Hypothesis Tests: Proportions
Related Practice
Textbook Question

Testing the Difference Between Two Means (c) calculate d̄ and Sd, Assume the samples are random and dependent, and the populations are normally distributed.

Interval Training

A researcher claims that sprint interval training improves running performance in trained athletes. The table shows the maximum aerobic speed (MAS), in kilometers per hour, of trained athletes before and after six sessions of sprint interval training. At , α=0.10 is there enough evidence to support the researcher’s claim? (Adapted from National Strength and Conditioning Association)

Textbook Question

Testing the Difference Between Two Means (b) find the critical value(s) and identify the rejection region(s), Assume the samples are random and dependent, and the populations are normally distributed.

Interval Training

A researcher claims that sprint interval training improves running performance in trained athletes. The table shows the maximum aerobic speed (MAS), in kilometers per hour, of trained athletes before and after six sessions of sprint interval training. At , α=0.10 is there enough evidence to support the researcher’s claim? (Adapted from National Strength and Conditioning Association)

Textbook Question

Testing the Difference Between Two Means (c) calculate d̄ and Sd, Assume the samples are random and dependent, and the populations are normally distributed.

[APPLET] Passing Play Percentages The passing play percentages of 10 randomly selected NCAA Division 1A college football teams for home and away games in the 2020–2021 season are shown in the table. At , α=0.20 is there enough evidence to support the claim that passing play percentage is different for home and away games? (Source: TeamRankings)

Textbook Question

Testing the Difference Between Two Means (d) find the standardized test statistic t, Assume the samples are random and dependent, and the populations are normally distributed.

Interval Training

A researcher claims that sprint interval training improves running performance in trained athletes. The table shows the maximum aerobic speed (MAS), in kilometers per hour, of trained athletes before and after six sessions of sprint interval training. At , α=0.10 is there enough evidence to support the researcher’s claim? (Adapted from National Strength and Conditioning Association)

Textbook Question

Testing the Difference Between Two Means (c) calculate d̄ and Sd, Assume the samples are random and dependent, and the populations are normally distributed.

[APPLET] Migraines

A researcher claims that injections of onabotulinumtoxinA reduce the number of days per month that chronic migraine sufferers have headaches. The table shows the number of days chronic migraine sufferers suffered migraines before and after using the treatment. At , α= 0.01 is there enough evidence to support the researcher’s claim? (Adapted from Journal of Headache and Pain)

Textbook Question

Testing the Difference Between Two Means, 

(d) decide whether to reject or fail to reject the null hypothesis. Assume the samples are random and independent, and the populations are normally distributed.

Transactions

 A magazine claims that the mean amount spent by a customer at Burger Stop is greater than the mean amount spent by a customer at Fry World. The results for samples of customer transactions for the two fast food restaurants are shown at the left. At , α=0.05 can you support the magazine’s claim? Assume the population variances are equal.

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