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The football coach wants to test if a protein shake meal supplement will improve the performance of their team over a 6 week period. 16 football players each drink a protein shake in the morning and in the evening every day for the 6 week period. They each have to complete a 100m sprint both before and after the 6 weeks. The difference in their running speed, in seconds, is computed as SPDAFT - SPDBEF, and the coach wants to see if the protein shakes have made the team faster at running the 100m. The sample mean difference from the 16 students was 0.1 seconds and the sample standard deviations was 0.13 seconds. Perform a test at the a = 0.10 level. (a) Set up the null and alternative hypothesis (using mathematical notation/numbers AND interpret them in context of the problem).

The football coach wants to test if a protein shake meal supplement will improve the performance of their team over a 6 week period. 16 football players each drink a protein shake in the morning and in the evening every day for the 6 week period. They each have to complete a 100m sprint both before and after the 6 weeks. The difference in their running speed, in seconds, is computed as SPDAFT - SPDBEF, and the coach wants to see if the protein shakes have made the team faster at running the 100m. The sample mean difference from the 16 students was 0.1 seconds and the sample standard deviations was 0.13 seconds. Perform a test at the a = 0.10 level. (a) Set up the null and alternative hypothesis (using mathematical notation/numbers AND interpret them in context of the problem).

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**Testing the Effectiveness of Protein Shake Supplements on Athletic Performance** The football coach wants to test if a protein shake meal supplement will improve the performance of their team over a 6-week period. Sixteen football players each drink a protein shake in the morning and in the evening every day for the 6-week period. They each have to complete a 100m sprint both before and after the 6 weeks. The difference in their running speed, in seconds, is computed as \(SPD_{AFT} - SPD_{BEF}\), and the coach wants to see if the protein shakes have made the team faster at running the 100m. The sample mean difference from the 16 students was 0.1 seconds and the sample standard deviation was 0.13 seconds. Perform a test at the \(\alpha = 0.10\) level. ### (a) Setting Up the Null and Alternative Hypotheses Null hypothesis (\(H_0\)): The protein shake supplements have no effect on the running speed. \[ H_0: \mu = 0 \] Alternative hypothesis (\(H_1\)): The protein shake supplements improve the running speed. \[ H_1: \mu > 0 \] ### (b) Calculate the Test Statistic The test statistic for a one-sample t-test is calculated as follows: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] Where: - \(\bar{x} = 0.1\) (sample mean difference) - \(\mu_0 = 0\) (mean difference under the null hypothesis) - \(s = 0.13\) (sample standard deviation) - \(n = 16\) (sample size) ### (c) Calculate the Critical Value For a one-tailed t-test at the \(\alpha = 0.10\) level with \(n - 1 = 15\) degrees of freedom, find the critical value from the t-distribution table. Interpret the results in the context of the problem and determine whether to reject the null hypothesis or not based on the calculated test statistic and critical value. This will help determine if the protein shake supplements have made the football team faster at running the 100m.
Transcribed Image Text:**Testing the Effectiveness of Protein Shake Supplements on Athletic Performance** The football coach wants to test if a protein shake meal supplement will improve the performance of their team over a 6-week period. Sixteen football players each drink a protein shake in the morning and in the evening every day for the 6-week period. They each have to complete a 100m sprint both before and after the 6 weeks. The difference in their running speed, in seconds, is computed as \(SPD_{AFT} - SPD_{BEF}\), and the coach wants to see if the protein shakes have made the team faster at running the 100m. The sample mean difference from the 16 students was 0.1 seconds and the sample standard deviation was 0.13 seconds. Perform a test at the \(\alpha = 0.10\) level. ### (a) Setting Up the Null and Alternative Hypotheses Null hypothesis (\(H_0\)): The protein shake supplements have no effect on the running speed. \[ H_0: \mu = 0 \] Alternative hypothesis (\(H_1\)): The protein shake supplements improve the running speed. \[ H_1: \mu > 0 \] ### (b) Calculate the Test Statistic The test statistic for a one-sample t-test is calculated as follows: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] Where: - \(\bar{x} = 0.1\) (sample mean difference) - \(\mu_0 = 0\) (mean difference under the null hypothesis) - \(s = 0.13\) (sample standard deviation) - \(n = 16\) (sample size) ### (c) Calculate the Critical Value For a one-tailed t-test at the \(\alpha = 0.10\) level with \(n - 1 = 15\) degrees of freedom, find the critical value from the t-distribution table. Interpret the results in the context of the problem and determine whether to reject the null hypothesis or not based on the calculated test statistic and critical value. This will help determine if the protein shake supplements have made the football team faster at running the 100m.
### Statistical Decision-Making and Test Statistic Interpretation #### Problem Description (d) Draw a picture of the distribution of the test statistic under \( H_0 \). Label and provide values for the critical value and the test statistic, and shade the critical region. (e) Make and justify a statistical decision at the \(\alpha = 0.10\) level and state your conclusions in context of the problem. --- ##### Detailed Instructions: **Diagram Explanation:** 1. **Distribution of the Test Statistic:** - The distribution under the null hypothesis \( H_0 \) is typically a standard normal distribution (or another appropriate distribution)). - Plot this distribution, generally centered around a mean value of 0. 2. **Critical Value:** - Identify the critical values corresponding to the chosen significance level (\(\alpha = 0.10\)). For a two-tailed test, each tail will contain \(0.05\) of the total area. - Label these points on the x-axis as the critical values. They are the cut-off points where we decide whether to reject \( H_0 \). 3. **Test Statistic:** - Calculate and mark the test statistic value on the x-axis. - Depending on whether it falls within one of the critical regions, you will decide whether to reject or fail to reject \( H_0 \). 4. **Shaded Critical Regions:** - Shade the regions under the distribution curve beyond the critical values. These regions represent the conditions under which \( H_0 \) would be rejected. **Statistical Decision:** 1. **Compare Test Statistic to Critical Values:** - If the test statistic falls within the shaded critical region, reject \( H_0 \). - If the test statistic falls outside the shaded critical region, fail to reject \( H_0 \). 2. **Justify the Decision:** - Based on the test statistic's position relative to the critical values, provide a rationale for rejecting or failing to reject \( H_0 \). - State the conclusion in the context of the original problem. **Example Conclusion:** - "At the \(\alpha = 0.10\) significance level, we reject \( H_0 \) because the test statistic falls within the critical region. This implies that there is sufficient evidence to support the alternative hypothesis \( H_1 \)." --- #### Explanatory Note: This educational content helps
Transcribed Image Text:### Statistical Decision-Making and Test Statistic Interpretation #### Problem Description (d) Draw a picture of the distribution of the test statistic under \( H_0 \). Label and provide values for the critical value and the test statistic, and shade the critical region. (e) Make and justify a statistical decision at the \(\alpha = 0.10\) level and state your conclusions in context of the problem. --- ##### Detailed Instructions: **Diagram Explanation:** 1. **Distribution of the Test Statistic:** - The distribution under the null hypothesis \( H_0 \) is typically a standard normal distribution (or another appropriate distribution)). - Plot this distribution, generally centered around a mean value of 0. 2. **Critical Value:** - Identify the critical values corresponding to the chosen significance level (\(\alpha = 0.10\)). For a two-tailed test, each tail will contain \(0.05\) of the total area. - Label these points on the x-axis as the critical values. They are the cut-off points where we decide whether to reject \( H_0 \). 3. **Test Statistic:** - Calculate and mark the test statistic value on the x-axis. - Depending on whether it falls within one of the critical regions, you will decide whether to reject or fail to reject \( H_0 \). 4. **Shaded Critical Regions:** - Shade the regions under the distribution curve beyond the critical values. These regions represent the conditions under which \( H_0 \) would be rejected. **Statistical Decision:** 1. **Compare Test Statistic to Critical Values:** - If the test statistic falls within the shaded critical region, reject \( H_0 \). - If the test statistic falls outside the shaded critical region, fail to reject \( H_0 \). 2. **Justify the Decision:** - Based on the test statistic's position relative to the critical values, provide a rationale for rejecting or failing to reject \( H_0 \). - State the conclusion in the context of the original problem. **Example Conclusion:** - "At the \(\alpha = 0.10\) significance level, we reject \( H_0 \) because the test statistic falls within the critical region. This implies that there is sufficient evidence to support the alternative hypothesis \( H_1 \)." --- #### Explanatory Note: This educational content helps
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