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

Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problem 8.3.12d

Chapter 8, Problem 8.3.12d

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)

Verified step by step guidance

Step 1: Calculate the differences (d) between the MAS before training and MAS after training for each athlete. For example, for Athlete 1, the difference is 18.5 - 18.5 = 0. Repeat this for all athletes.

Step 2: Compute the mean of the differences (d̄). Add all the differences calculated in Step 1 and divide by the number of athletes (n = 8). Use the formula: d̄ = (Σd) / n.

Step 3: Calculate the standard deviation of the differences (sd). Use the formula: sd = sqrt((Σ(d - d̄)^2) / (n - 1)), where d̄ is the mean difference from Step 2.

Step 4: Compute the standardized test statistic t using the formula: t = d̄ / (sd / sqrt(n)), where d̄ is the mean difference, sd is the standard deviation of the differences, and n is the number of athletes.

Step 5: Compare the calculated t-value to the critical t-value at α = 0.10 for a two-tailed test with degrees of freedom df = n - 1 (df = 7). If the calculated t-value exceeds the critical t-value, there is enough evidence to support the researcher’s claim.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 same subjects are measured under different conditions. In this case, the maximum aerobic speed (MAS) of athletes is measured before and after sprint interval training. This relationship allows for a direct comparison of the effects of the training on the same individuals, which can enhance the statistical power of the analysis.

Standardized Test Statistic (t)

The standardized test statistic, commonly denoted as t, is used to determine if there is a significant difference between the means of two groups. In this context, it helps assess whether the change in MAS before and after training is statistically significant. The t-statistic is calculated by taking the difference between the sample means, dividing it by the standard error of the difference, and is compared against a critical value from the t-distribution.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this scenario, the null hypothesis (H0) posits that there is no difference in MAS before and after training, while the alternative hypothesis (H1) suggests that there is a significant improvement. The significance level (α) of 0.10 indicates the threshold for rejecting the null hypothesis, guiding the researcher in determining if the evidence supports the claim of improved performance.

Recommended video:

Guided course

06:21

Step 1: Write Hypotheses

Related Practice

Textbook Question

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.

views

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)

views

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)

views

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

views

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

views

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] 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)

views