Question

At Rachel's 11^th birthday party, 8 girls were timed to see how long (in seconds) they could hold their breath in a relaxed position. After a two-minute rest, they timed themselves while jumping. The girls thought that the mean difference between their jumping and relaxed times would be zero. Test their hypothesis at the 5% level.

Relaxed time (seconds)	Jumping time (seconds)
29	21
47	42
32	26
22	21
23	25
45	43
37	35
29	32

NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

• Part (e) What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer to three decimal places.)
  

• Part (f)

  What is the p-value? (Round your answer to four decimal places.)
  

Answer 1

The hypothesis to be tested is Ho : Mu d=0 (Average difference)

Ha :Mu d is not equal to zero

The sample statistic that will be used is

t= average difference from sample - mu d/ Sd/n^1/2  This is a t variable with n-1 degrees of freedom

n=8 hence degree of freedom = 8-1 =7 level of significance is 5% (two tailed) ie 2.5%on both sides of the t curve.

The data is computed as follows :

Jumping Time (seconds)	Difference	Difference-Mean	difference - mean square
21	8	5.625	31.640625
42	5	2.625	6.890625
26	6	3.625	13.140625
21	1	-1.375	1.890625
25	-2	-4.375	19.140625
43	2	-0.375	0.140625
35	2	-0.375	0.140625
32	-3	-5.375	28.890625
	19		101.875
Average difference	2.375
Variance of differences	14.553571

Hence sample mean difference = 2.375

Sample variance = 14.55 (please note df is 7 and not 8 while computing this)

n= 8

putting all of this in the t statistic we get

t= 8^1/2 * 2.375 -0 (assuming Ho to be true) / 3.51

on solving we get t= 1.913

The p value of getting this value when degree of freedom is 7 and alpha 5% (two tailed) is 0.0973

As this is greater than 5% we do not reject the nulll hypothesis. Hence we conclude that that the mean difference in the two times in not significantly different from 0.