Question

A researcher recruited a sample of 5 women who were interested in trying a new six-week program of diet and exercise designed to promote healthy weight loss. She plans to use the .01 significance level to test whether weight decreases (on average) for women using this program. The table below gives the starting and end weights (in pounds) for each of the five women.

Subject	Starting Wt. (Sample 1)	Ending Wt. (Sample 2)
1	174	156
2	122	110
3	169	160
4	181	165
5	147	139

a. Should these samples be considered paired or independent? Why?

b. Chose the appropriate formula for the test statistic and finds its value.

c. Describe the rejection region for this test.

d. What should the researcher conclude?

e. Find a 95% confidence interval for the mean weight-loss of women using this program. What mean weightless would you predict for women who follow this program?

An independently selected sample of five men also participated in the same study. The table below shows results for the number of pounds lost by the five men and the five women in the study. The researcher will use the .01 significance level to test whether (on average) the program produces different weight loss results for men and women. You may assume the population variances are equal (although the sample variances are not).

	Weight Loss (in pounds)
	Men (Sample 1)	Women (Sample 2)
Sample Size	5	5
Sample mean	19.2	12.6
Standard deviation	4.970	4.336

f. Formulate the hypothesis for this test.

g. Should the pooled-sample variance be used in this situation? Why?

h. Choose the appropriate formula for the test statistic and find its value.

i. What is the rejection region for this test?

j. What should the researcher conclude?

Answer 1

Solution:-

a) These samples should be considered pairred.

b) State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u_d< 0

Alternative hypothesis: u_d > 0

Note that these hypotheses constitute a right-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a matched-pairs t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard deviation of the differences (s), the standard error (SE) of the mean difference, the degrees of freedom (DF), and the t statistic test statistic (t).

s = sqrt [ (\sum (d_i - d)² / (n - 1) ]

s = 4.3359

SE = s / sqrt(n)

S.E = 1.9391

DF = n - 1 = 5 -1

D.F = 4

t = [ (x₁ - x₂) - D ] / SE

t = 6.498

c)

t_critical = 3.747

Rejection region is t > 3.747.

where d_i is the observed difference for pair i, d is mean difference between sample pairs, D is the hypothesized mean difference between population pairs, and n is the number of pairs.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 4 degrees of freedom is more extreme than 6.498; that is, less than - 6.498 or greater than 6.498.

Thus, the P-value = less than 0.001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

d) From the above test we have sufficient evidence in the favor of the claim that weight decreases (on average) for women using this program.