91Ó°ÊÓ

The study described in the previous exercise also measured time to exhaustion for the 11 triathletes on a day when they listened to music that the runners had classified as neutral as compared to motivational. The researchers calculated the difference between the time to exhaustion while running to motivational music and while running to neutral music. The mean difference in (motivational - neutral) was -7 seconds (the sample mean time to exhaustion was actually lower when listening to music the runner viewed as motivational than the mean when listening to music the runner viewed as neutral). Suppose that the standard deviation of the differences was \(s_{d}=80 .\) For purposes of this exercise, assume that it is reasonable to regard these 11 triathletes as representative of the population of experienced triathletes and that the population difference distribution is approximately normal. Is there convincing evidence that the mean time to exhaustion for experienced triathletes running to motivational music differs from the mean time to exhaustion when running to neutral music? Carry out a hypothesis test using \(\alpha=0.05 .\)

Step by step solution

01

1. State the hypotheses:

H0 (null hypothesis): The mean difference between the time to exhaustion when running to motivational music and when running to neutral music is equal to 0. Mathematically: μd = 0 H1 (alternative hypothesis): The mean difference between the time to exhaustion when running to motivational music and when running to neutral music is not equal to 0. Mathematically: μd ≠ 0

02

2. Calculate degrees of freedom:

The degrees of freedom (df) in this case is the sample size (n) minus 1. df = n - 1 = 11 - 1 = 10

03

3. Determine the t-critical values:

Using a t-table and the given α=0.05, we can find the t-critical values for a two-tailed test and df=10. t-critical = ±2.228

04

4. Compute the t-statistic:

The t-statistic can be calculated using the following formula: t = (sample mean difference - population mean difference) / (standard deviation of differences / sqrt(sample size)) t = (-7 - 0) / (80 / sqrt(11)) = -7 / (80 / sqrt(11)) ≈ -0.598

05

5. Compare the t-statistic to the t-critical values:

Now, we will compare the t-statistic to the t-critical values to determine if there is convincing evidence to reject the null hypothesis. (-2.228) < -0.598 < (2.228) Since the calculated t-statistic falls within the range of the t-critical values, we fail to reject the null hypothesis.

06

6. Conclusion

There is not enough evidence at the α=0.05 level to conclude that there is a significant difference between the mean time to exhaustion for experienced triathletes running to motivational music and when running to neutral music. The test result is not statistically significant, and we fail to reject the null hypothesis.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

Understanding the null hypothesis is the cornerstone of hypothesis testing in statistics. It proposes that there is no effect or no difference, and in the context of our triathlete study on motivational versus neutral music, the null hypothesis asserts that there is no difference in time to exhaustion. Mathematically, it is expressed as \( H_0: \mu_d = 0 \), with \( \mu_d \) representing the population mean difference. The null hypothesis serves as the default position that a researcher aims to challenge; it is presumed true until evidence indicates otherwise.
Let's consider this null hypothesis against our real-world scenario: it suggests that for the wider population of triathletes, motivational music does not influence the time to exhaustion when compared to neutral music.

Alternative Hypothesis

While the null hypothesis predicts no effect, the alternative hypothesis \( H_1 \) represents the opposite. It posits that there is an effect or a difference. In our triathlon study, the alternative hypothesis states that there is a difference in time to exhaustion when listening to motivational music as opposed to neutral music, and is expressed as \( H_1: \mu_d eq 0 \).
The research is designed to provide evidence that will support or refute the alternative hypothesis. In hypothesis testing, if the null hypothesis can be rejected, then the alternative hypothesis is said to have merit. In other cases, if the evidence isn't strong enough, we may retain the null hypothesis as in the case of our exercise.

T-Critical Values

Understanding T-Critical Values

T-critical values, also known as critical t-values, are predetermined points on the t-distribution. These values help in determining the threshold at which the null hypothesis can be rejected. The critical values depend on the desired level of significance \( \alpha \) and the degrees of freedom \( df \) which is based on the sample size. For our triathlon example with \( \alpha = 0.05 \) and \( df = 10 \) from a sample of 11 triathletes, the t-critical values for a two-tailed test are \( \pm2.228 \).
These values act as cutoff points: if the calculated t-statistic falls beyond the critical values, we have grounds to reject the null hypothesis. However, if the t-statistic lies within the critical values, as it does in our scenario, we do not have sufficient evidence to do so.

T-Statistic

Role of the T-Statistic

The t-statistic is a central component in hypothesis testing; it's a calculated value that measures the degree of difference between the sample mean and the population mean as stipulated by the null hypothesis. To calculate the t-statistic for our example, one uses the formula: \[ t = \frac{(\text{sample mean difference} - \text{population mean difference})}{(\text{standard deviation of differences} / \sqrt{\text{sample size}})} \]. The resulting t-statistic allows us to compare the observed sample data against the null hypothesis. Our calculated t-statistic was approximately -0.598, revealing that the observed difference in times is not strong enough to suggest a significant effect of motivational music on the time to exhaustion for triathletes.

Statistical Significance

Determining Statistical Significance

Statistical significance indicates the likelihood that the result obtained is due to something other than random chance. In hypothesis testing, we use a significance level (denoted as \( \alpha \) ), typically set at 0.05 or 5%. If our t-statistic falls outside the boundary of the t-critical values, we would conclude that our results are statistically significant, leading to the rejection of the null hypothesis.
In the exercise with our triathletes, because our calculated t-statistic did not exceed the t-critical values, our results are not statistically significant. Thus, we retain the null hypothesis indicating that the data do not provide strong enough evidence to conclude a difference in time to exhaustion when listening to motivational versus neutral music in the context of experienced triathletes.