Question

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

Population variances are equal. This is an independent t test

f) NULL HPOTHESIS H0:

ALTERNATIVE HYPOTHESIS HA:

Alpha= 0.01

g) Yes,the pooled-sample variance should be used in this situation. Because population variances are equal.

h) Under null hypothesis test statistic is

where xbar is sample mean for men , ybar is sample mean for women and Sp is pooled standard deviation.

s_p^2 = \frac{(n_1-1) s_1^2 + (n_2-1) s_2^2}{n_1+n_2-2}sp2​=n1​+n2​−2(n1​−1)s12​+(n2​−1)s22​​

Plugging in the corresponding values in the formula above, we get that

sp^2​=(5−1)(4.97)^2+(5−1)(4.336)^2​/(5+5−2)

sp^2=21.7509

Now that we have the pooled variance, we can compute the pooled standard deviation by simply taking the squared root of the value obtained for the variance.

The following pooled standard deviation is obtained:

sp​=sqrt(21.7509)​=4.6638

d.f= 5+5-2 8

i) Rejection region if t> t critical( 3.35) with d.f=8 and at alpha=0.01

j) Conclusion: Since calculated value of t is smaller than critical value of t therefore we DO NOT REJECT NULL HYPOTHESIS. We do not have sufficient evidence to show that the program produces different weight loss results for men and women.