Question

A fitness course claims that it can improve an individual's physical ability. To test the effect of a physical fitness course on one's physical ability, the number of sit-ups that a person could do in one minute, both before and after the course, was recorded. Ten individuals are randomly selected to participate in the course. The results are displayed in the following table. Can it be concluded, from the data, that participation in the physical fitness course resulted in significant improvement? Let d=(number of sit-ups that can be done after taking the course)−(number of sit-ups that can be done prior to taking the course). Use a significance level of α=0.1 for the test. Assume that the numbers of sit-ups are normally distributed for the population both before and after taking the fitness course.

Sit-ups before 44 33 40 32 21 35 52 24 32 48

Sit-ups after 58 36 53 40 37 39 58 38 36 60

Step 1 of 5: State the null and alternative hypotheses for the test.

Step 2 of 5:

Find the value of the standard deviation of the paired differences. Round your answer to one decimal place.

Step 3 of 5:

Compute the value of the test statistic. Round your answer to three decimal places.

Step 4 of 5:

Determine the decision rule for rejecting the null hypothesis H0H0. Round the numerical portion of your answer to three decimal places.

Step 5 of 5:

Make the decision for the hypothesis test.

Answer 1

Step 1

To Test :-

H0 :- µd = 0

H1 :- µd > 0

Step 2

S(d) = √(Σ (di - d̅)2 / n-1)
S(d) = √(218.4 / 10-1) = 4.9

Step 3

Test Statistic :-
S(d) = √(Σ (di - d̅)2 / n-1)
S(d) = √(218.4 / 10-1) = 4.9261
d̅ = Σ di/n = 94 / 10 = 9.4
t = d̅ / ( S(d) / √(n) )
t = 9.4 / ( 4.9261 / √(10) )
t = 6.034

Step 4

Test Criteria :-
Reject null hypothesis if t > t(α)
Critical value t(α) = t(0.1) = 1.383
t > t(α) = 6.0342 > 1.383
Result :- Reject null hypothesis

Step 5

There is sufficient evidence to support the claim  that it can improve an individual's physical ability.