Researchers wish to study the number of minutes it takes for amateur athletes to

Question

Researchers wish to study the number of minutes it takes for amateur athletes to run a mile with and

without a special nutrition drink. The running times for a sample of 10 athletes are listed below:

ID with without

1: 9.4; 11.4

2: 9.8; 10.9

3: 11.6; 12.7

4: 10.1; 10.9

5: 10.1; 10.3

6: 11.7; 14.3

7: 10.5; 11.7

8: 8.7; 7.5

9: 9.3; 10.8

10: 9.6; 9.8

Summary statistics:

Mean time with drink: 10.08

Mean time without drink: 11.03

Sample std. dev. with drink: 0.97

Sample std. dev. without drink: 1.78

a)

Use a paired t-test to test whether the difference between the two groups of readings (with and without

the drink). What is the p-value?

b) Repeat the test using a regular two sample t-test. What is the p-value?

c) Briefly, explain why the results are different.

Explanation / Answer

a) Here I have used r programming to solve the problem.

Here p value is 0.01911 i.e. less than alpha =0.05

with = c(9.4, 9.8, 11.6, 10.1, 10.1, 11.7, 10.5, 8.7, 9.3, 9.6)
without = c( 11.4,10.9,12.7, 10.9, 10.3, 14.3, 11.7, 7.5, 10.8, 9.8)
length(with)
length(without)
t.test(with,without, paired=TRUE)

Hence we reject the null hypothesis.

b) Regular two sample t test. Here p value is 0.156

Here p value is 0.156 i.e. greater than alpha =0.05. Hence we accept the null hypothesis.

t.test(with,without, var.equal=TRUE, paired=FALSE)

c) The results are different because in the 1st case we are considering the observations are paired i.e. correlated . While in the 2nd case we are considering the observations are independent i.e. uncorrelated. This is why the esults are different.

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