Question

A new exercise program is proposed to increase the V02 max (a measure of fitness in mL/kg/min) of men with sedentary lifestyles. The researchers randomly recruited 25 such men from their local health care centre. They measured the V02 max of each individual at the start of the study (pre) and then again after 6 months of participating in the exercise program (post). The data are available in the file VO2max.csv.

a. What are the null and alternative hypotheses?

b. (1 mark) What is the value of the test statistic?

c. (1 mark) What is the distribution of the test statistic if the null hypothesis is true?

d. What is the p-value?

e. Using a significance level of 5%, state your conclusions about this test in the language of the problem.

f. Name the two key unstated assumptions required for the validity of your hypothesis test and explain why they are reasonable in this context.

The data are available in the file VO2max.csv.

"pre","post"

28.3,28.9

29.3,36.5

34.7,34.5

30.2,30.6

30.4,37.8

35.1,35.3

31.4,33.1

26.2,36.7

27.9,36.6

28.7,36.5

33.7,36.1

31.1,35.7

31.2,33.8

30.3,33.1

28.3,32.9

35.4,31.9

31.5,33.4

24.1,30.2

32.1,40.5

28.6,37.6

26.8,30.6

29.3,32.8

26.9,32.6

27.8,36.3

28.1,33.7

Answer 1

Here we have paired data so paired t test will be used. Let d = pre -post

Following table shows the calculations:

	pre	post	d	(d-mean)^2
	28.3	28.9	-0.6	14.5161
	29.3	36.5	-7.2	7.7841
	34.7	34.5	0.2	21.2521
	30.2	30.6	-0.4	16.0801
	30.4	37.8	-7.4	8.9401
	35.1	35.3	-0.2	17.7241
	31.4	33.1	-1.7	7.3441
	26.2	36.7	-10.5	37.0881
	27.9	36.6	-8.7	18.4041
	28.7	36.5	-7.8	11.4921
	33.7	36.1	-2.4	4.0401
	31.1	35.7	-4.6	0.0361
	31.2	33.8	-2.6	3.2761
	30.3	33.1	-2.8	2.5921
	28.3	32.9	-4.6	0.0361
	35.4	31.9	3.5	62.5681
	31.5	33.4	-1.9	6.3001
	24.1	30.2	-6.1	2.8561
	32.1	40.5	-8.4	15.9201
	28.6	37.6	-9	21.0681
	26.8	30.6	-3.8	0.3721
	29.3	32.8	-3.5	0.8281
	26.9	32.6	-5.7	1.6641
	27.8	36.3	-8.5	16.7281
	28.1	33.7	-5.6	1.4161
	Total		-110.3	300.3265

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(c) (1 mark) What is the distribution of the test statistic if the null hypothesis is true?

The distribution will be t distribution with df=24.

(e)

There is evidence to conclude that new exercise program increases the V02 max (a measure of fitness in mL/kg/min) of men with sedentary lifestyles.

(f)

The variable are approximately normally distributed. And observations are independent from other.

Since sample size is large and participants are independent from each other so assumptions seems to be reasonable.