Question

Hi, I am taking stats class and I have one question. I have been struggling with it for few hours.

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_d = μ_water − μ_{guar syrup}. Round your test statistic to two decimal places and the P-value to three decimal places.)

Swimmer	Velocity (m/s)
	Water	Guar Syrup
1	1.10	1.12
2	1.81	0.91
3	1.20	1.83
4	0.93	1.92
5	1.20	1.38
6	1.31	1.59
7	1.00	1.92
8	1.04	1.00
9	1.94	1.56
10	1.02	1.60
11	1.20	1.65
12	0.95	1.85
13	1.72	1.96
14	1.97	1.59
15	0.97	1.60
16	1.02	0.94
17	0.91	1.39
18	1.34	1.55
19	1.99	1.39
20	1.65	1.10

t	=
df	=
P	=

Is there sufficient evidence to suggest that there is any difference in swimming time between swimming in guar syrup and swimming in water? Carry out a hypothesis test using α = 0.01 significance level.

YesNo    

Answer 1

SN	Water	Guar Syrup	d	d2
1	1.1	1.12	-0.02	0.0004
2	1.81	0.91	0.9	0.81
3	1.2	1.83	-0.63	0.3969
4	0.93	1.92	-0.99	0.9801
5	1.2	1.38	-0.18	0.0324
6	1.31	1.59	-0.28	0.0784
7	1	1.92	-0.92	0.8464
8	1.04	1	0.04	0.0016
9	1.94	1.56	0.38	0.1444
10	1.02	1.6	-0.58	0.3364
11	1.2	1.65	-0.45	0.2025
12	0.95	1.85	-0.9	0.81
13	1.72	1.96	-0.24	0.0576
14	1.97	1.59	0.38	0.1444
15	0.97	1.6	-0.63	0.3969
16	1.02	0.94	0.08	0.0064
17	0.91	1.39	-0.48	0.2304
18	1.34	1.55	-0.21	0.0441
19	1.99	1.39	0.6	0.36
20	1.65	1.1	0.55	0.3025
		Total	-3.58	6.1818

We have to test

H₀: µ_water - µ_gaur =0 i.e. μ_d = 0

H_A: µ_water - µ_gaur ≠ 0 i.e. μ_d ≠ 0

We have got

n=20

Σd=-3.58

Σd²=6.1818

The test statistic to test this null hypothesis is paired t-test given as below:

The test statistic = t =-1.48

The degrees of freedom = df = n-1 = 19

The p-value for above test = 0.155

The p-value of above test is 0.155 which is greater than α=0.01 and it suggests that we do not have enough evidence against null hypothesis to reject it, so we fail to reject the null hypothesis and we can conclude that there is no significant difference in swimming time between swimming in guar syrup and swimming in water.

In other words, there is not sufficient evidence to suggest that there is any difference in swimming time between swimming in guar syrup and swimming in water.