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 the table below.

Swimmer	Velocity (m/s)
Water	Guar Syrup
1	0.90	0.94
2	0.92	0.97
3	1.00	0.95
4	1.10	1.15
5	1.20	1.24
6	1.25	1.24
7	1.25	1.27
8	1.30	1.30
9	1.35	1.33
10	1.40	1.42
11	1.40	1.43
12	1.50	1.54
13	1.65	1.58
14	1.70	1.70
15	1.75	1.80
16	1.80	1.77
17	1.80	1.83
18	1.85	1.86
19	1.90	1.88
20	1.95	1.95

The researchers concluded that swimming in guar syrup does not change mean swimming speed. Are the given data consistent with this conclusion? Carry out a hypothesis test using a 0.01 significance level. (Use

μ_d = μ_water − μ_{guar syrup}.)

State the appropriate null and alternative hypotheses.

H₀: μ_d ≠ 0

H_a: μ_d = 0

H₀: μ_d = 0

H_a: μ_d ≠ 0

    

H₀: μ_d < 0

H_a: μ_d = 0

H₀: μ_d = 0

H_a: μ_d > 0

H₀: μ_d = 0

H_a: μ_d < 0

Find the test statistic and P-value. (Round your test statistic to one decimal place and your P-value to three decimal places.)

t=

P-value=

State the conclusion in the problem context.

We fail to reject H₀. The data provide convincing evidence that swimming in guar syrup changes mean swimming speed.

We reject H₀. The data provide convincing evidence that swimming in guar syrup changes mean swimming speed.   

We reject H₀. The data do not provide convincing evidence that swimming in guar syrup changes mean swimming speed.

We fail to reject H₀. The data do not provide convincing evidence that swimming in guar syrup changes mean swimming speed

You may need to use the appropriate table in the appendix or technology to answer this question.

Answer 1

Using Excel, go to Data, select Data Analysis, choose t-Test: Paired Two Sample for Means at Alpha = 0.01.

	Water	Gaur Syrup
Mean	1.449	1.458
Variance	0.113	0.108
Observations	20	20
Pearson Correlation	0.995
Hypothesized Mean Difference	0
df	19
t Stat	-1.168
P(T<=t) one-tail	0.129
t Critical one-tail	2.539
P(T<=t) two-tail	0.257
t Critical two-tail	2.861

H₀: μ_d = 0

H_a: μ_d ≠ 0

t statistic = -1.2

p-value = 0.257

We fail to reject H₀. The data do not provide convincing evidence that swimming in guar syrup changes mean swimming speed.