Question

David was curious if regular excise really helps weight loss, hence he decided to perform a hypothesis test. A random sample of 5 UMUC students was chosen. The students took a 30- minute exercise every day for 6 months. The weight was recorded for each individual before and after the exercise regimen. Does the data below suggest that the regular exercise helps weight loss? Assume David wants to use a 0.05 significance level to test the claim.

(a) What is the appropriate hypothesis test to use for this analysis: z-test for two proportions, t-test for two proportions, t-test for two dependent samples (matched pairs), or t-test for two independent samples? Please identify and explain why it is appropriate.

(b) Let μ1 = mean weight before the exercise regime. Let μ2 = mean weight after theexercise regime. Which of the following statements correctly defines the null hypothesis?

(i) μ1 - μ2 > 0 (μd > 0) (ii) μ1 - μ2 = 0 (μd = 0) (iii) μ1 - μ2 < 0 (μd < 0)

(c) Let μ1 = mean weight before the exercise regime. Let μ2 = mean weight after theexercise regime. Which of the following statements correctly defines the alternative hypothesis?

(a) μ1 - μ2 > 0 (μd > 0) (b) μ1 - μ2 = 0 (μd = 0) (c) μ1 - μ2 < 0 (μd < 0)

(d) Determine the test statistic. Round your answer to three decimal places. Show all work; writing the correct test statistic, without supporting work, will receive no credit.

(e) Determine the p-value. Round your answer to three decimal places. Show all work; writing the correct critical value, without supporting work, will receive no credit.

(f) Compare p-value and significance level α. What decision should be made regardingthe null hypothesis (e.g., reject or fail to reject) and why?

(g) Is there sufficient evidence to support the claim that regular exercise helps weight loss? Justify your conclusion.

Subject Before After

1 190 180

2 170 160

3 185 190

4 160 160

5 200 190

Answer 1

(a)

t-test for two dependent samples (matched pairs)

Since we are observing the weight of the same 5 UMUC students before and after following the 30 minutes exercise regime for 6 months, therefore the samples are dependent. Hence using a matched pair test is appropriate in this situation.

(b)

(ii) μ1 - μ2 = 0 (μd = 0)

The null hypothesis is the statement of no effect.

(c)

(a) μ1 - μ2 > 0 (μd > 0)

The alternative hypothesis would be testing whether there is a weight change by following the new regime i.e. whether the weight before was greater than the weight after.

(d)

The following table is obtained:

	Sample 1	Sample 2	Difference = Sample 1 - Sample 2
	190	180	10
	170	160	10
	185	190	-5
	160	160	0
	200	190	10
Average	181	176	5
St. Dev.	15.969	15.166	7.071
n	5	5	5

For the score differences, we have

The t-statistic is computed as shown in the following formula:

(e)

The p-value is given by

P(t>1.581) = p = 0.0945

(f)

Since , it is concluded that the null hypothesis is not rejected.

(g)

Since it is observed that , it is then concluded that the null hypothesis is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is greater than μ2​, at the 0.05 significance level. Hence there is not sufficient evidence to support the claim that regular exercise helps weight loss.

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