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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 8.3.18d
Chapter 8, Problem 8.3.18d

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.
[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)
Table displaying home and away passing play percentages for 10 NCAA Division 1A college football teams.

Verified step by step guidance
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Step 1: Calculate the differences (d) between the home and away passing play percentages for each college team. For each team, subtract the away percentage from the home percentage.
Step 2: Compute the mean of the differences (d̄). Add all the differences calculated in Step 1 and divide by the number of teams (n = 10).
Step 3: Calculate the standard deviation of the differences (s_d). Use the formula for standard deviation: s_d = sqrt((Σ(d_i - d̄)^2) / (n - 1)), where d_i represents each individual difference and d̄ is the mean difference.
Step 4: Compute the standardized test statistic t using the formula: t = (d̄) / (s_d / sqrt(n)), where d̄ is the mean difference, s_d is the standard deviation of the differences, and n is the number of teams.
Step 5: Compare the calculated t-value to the critical t-value at α = 0.20 for a two-tailed test with degrees of freedom df = n - 1 (df = 9). If the absolute value of the calculated t is greater than the critical t-value, reject the null hypothesis; otherwise, fail to 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.

Dependent Samples

Dependent samples, also known as paired samples, occur when the samples are related or matched in some way. In this context, the passing play percentages for home and away games are paired for the same teams, meaning that each team's performance at home is directly compared to its performance away. This relationship is crucial for applying statistical tests that account for the dependency between the samples.
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Standardized Test Statistic (t)

The standardized test statistic, denoted as t, is used to determine how far the sample mean difference is from the hypothesized population mean difference, measured in standard error units. In this scenario, it helps assess whether the observed differences in passing play percentages between home and away games are statistically significant. The calculation of t involves the means, standard deviations, and sample sizes of the paired samples.
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Step 2: Calculate Test Statistic

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this case, the null hypothesis would state that there is no difference in passing play percentages between home and away games, while the alternative hypothesis posits that a difference exists. The significance level (α = 0.20) indicates the threshold for rejecting the null hypothesis, guiding the decision-making process based on the calculated t statistic.
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Step 1: Write Hypotheses
Related Practice
Textbook Question

Testing the Difference Between Two Means (e) decide whether to reject or fail to reject the null hypothesis, 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) find the standardized test statistic t, 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) 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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Textbook Question

Testing the Difference Between Two Means (e) decide whether to reject or fail to reject the null hypothesis, 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)