Question

Twelve runners are asked to run a 10-kilometer race on each of two consecutive weeks. In
one of the races, the runners wear one brand of shoe and in the other a different brand. The
brand of shoe they wear in which race is determined at random. All runners are timed and
are asked to run their best in each race. The results (in minutes) are given below:
Runner Brand 1 Brand 2
1 31.23 32.02
2 29.33 28.98
3 30.50 30.63
4 32.20 32.67
5 33.08 32.95
6 31.52 31.53
7 30.68 30.83
8 31.05 31.10
9 33.00 33.12
10 29.67 29.50
11 30.55 30.57
12 32.12 32.20
Use the sign test for matched pairs to determine if there is evidence that times using Brand
1 tend to be faster than times using Brand 2.
(a) What are the hypotheses we wish to test?
i. H0 : μ = 0 versus Ha : μ > 0, where μ =the mean of the differences in running
times (Brand 2-Brand 1) for all runners who run this race twice wearing the two
brands of shoes.
ii. H0 : p = 1
2 versus Ha : p , 1
2 , where p = the proportion of running times using
Brand 1 that are faster than times using Brand 2.
iii. H0 : p = 1
2 versus Ha : p >
1
2 , where p = the proportion of running times using
Brand 1 that are faster than times using Brand 2.
iv. H0: population median =0 versus Ha: population median , 0, where the median
of the differences in running times for all runners who run this race twice wearing
the two brands of shoes is measured for Brand 2-Brand 1.
(b) What is the (approximate) value of the P-value?
(c) Determine which of the following statements is true.
i. We would not reject the null hypothesis of no difference at the 0.10 level.
ii. We would reject the null hypothesis of no difference at the 0.10 level but not at the
0.05 level.
iii. We would reject the null hypothesis of no difference at the 0.05 level but not at the
0.01 level.
iv. We would reject the null hypothesis of no difference at the 0.01 level.

Answer 1