Question

To investigate the fluid mechanics of swimming, twenty swimmers each swam a specified distance in a water-filled pool and in a pool where the water was thickened with food grade guar gum to create a syrup-like consistency. Velocity, in meters per second, was recorded and the results are given in a table below. The researchers concluded that swimming in guar syrup does not change swimming speed. (Use a statistical computer package to calculate the P-value. Use μ_water − μ_{guar syrup}. Round your test statistic to two decimal places and your P-value to three decimal places.)

Swimmer	Velocity (m/s)
	Water	Guar Syrup
1	1.02	1.07
2	1.31	1.81
3	1.81	1.41
4	0.90	1.61
5	0.94	1.57
6	1.23	1.17
7	1.61	1.54
8	1.13	1.36
9	1.26	0.90
10	1.88	1.63
11	1.54	1.76
12	1.51	1.26
13	1.88	1.89
14	1.96	1.54
15	1.74	1.55
16	0.93	1.61
17	1.68	1.26
18	1.49	1.79
19	1.34	1.71
20	1.24	1.81

t	=
df	=
P-value	=

Answer 1

Since all the 20 swimmers swam in water filled pool and guar syrup filled pool, these are paired (dependent) samples. Hence we will be conductiong the test for difference between means with dependent samples.

Let be the true mean swimming speeds in a pool filled with water and guar syprup respectively.

The researchers want to concluded that swimming in guar syrup does not change swimming speed, that is there is no difference between , or

We want to test the following hypotheses

Since there is only one sample (as the 2 observations are dependent) let be the true mean difference between the 2 speeds

Let d be the difference in the speeds of individual swimmer in the sample.

the difference is calculated as below

Swimmer	Velocity (m/s)
	Water	Guar Syrup	difference (d)
1	1.02	1.07	-0.05
2	1.31	1.81	-0.5
3	1.81	1.41	0.4
4	0.9	1.61	-0.71
5	0.94	1.57	-0.63
6	1.23	1.17	0.06
7	1.61	1.54	0.07
8	1.13	1.36	-0.23
9	1.26	0.9	0.36
10	1.88	1.63	0.25
11	1.54	1.76	-0.22
12	1.51	1.26	0.25
13	1.88	1.89	-0.01
14	1.96	1.54	0.42
15	1.74	1.55	0.19
16	0.93	1.61	-0.68
17	1.68	1.26	0.42
18	1.49	1.79	-0.3
19	1.34	1.71	-0.37
20	1.24	1.81	-0.57

the sample mean difference is

the sample standard deviation of difference is

Since we do not know the population standard deviation, we estimate the same using sample SD

the standard error of mean difference is

The hypothesized value of mean difference is zero, that is

The sample size n=20 is less than 30 and the population standard deviation is not known. Hence we will use t-distribution to test the hypotheses.

The test statistics is

The degrees of freedom df=(n-1) = (20-1) =19

this is a 2 tailed test ("" in the alternative hypothesis)

The p-value is the sum of area under rigth and left tail, which is P(T>1.06)+P(T<-1.06)

Using the excel function =T.DIST.2T(1.06,19) we get

p-value=0.302

Since the p-value is greater than the lvel of significance alpha =0.05, we can not reject the null hypothesis.

We conclude that there is no sufficient evidence to reject the claim that swimming in guar syrup does not change swimming speed.