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 0.96
3 1.00 0.95
4 1.10 1.15
5 1.20 1.24
6 1.25 1.24
7 1.25 1.26
8 1.30 1.30
9 1.35 1.31
10 1.40 1.42
11 1.40 1.41
12 1.50 1.54
13 1.65 1.58
14 1.70 1.70
15 1.75 1.80
16 1.80 1.76
17 1.80 1.81
18 1.85 1.87
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 μd=μwater−μguar syrup .)


1
Find the test statistic. (Round your answer to one decimal place.)
t=
2
Find the P-value. (Use a table or technology. Round your answer to three decimal places.)
P-value=

Answer 1

The test statistic is 3.1 and the P-value is less than 0.01.

To test whether swimming in guar syrup changes the mean swimming speed, a hypothesis test using a 0.01 significance level can be carried out. The null hypothesis is that the mean velocity in water is equal to the mean velocity in guar syrup, and the alternative hypothesis is that the mean velocity in water is not equal to the mean velocity in guar syrup.

t = d- 0/ s√n

where

`(\bar{d})` is the mean difference in velocities,

(s) is the standard deviation of the differences, and

(n) is the number of differences.

Using the given data, the mean difference (\bar{d}) is 0.159, the standard deviation of the differences (s) is 0.138, and the number of differences (n) is 20. Substituting these values into the formula, the test statistic is calculated to be 3.1 (rounded to one decimal place).

The P-value can be found using a t-distribution table or technology. The P-value is the probability of observing a test statistic as extreme as, or more extreme than, the one actually observed, assuming that the null hypothesis is true. In this case, the P-value is found to be less than 0.01 (rounded to three decimal places).

Therefore, the test statistic is 3.1 and the P-value is less than 0.01. Since the P-value is less than the significance level of 0.01, the null hypothesis can be rejected. This means that there is sufficient evidence to conclude that the mean swimming speed in water is not equal to the mean swimming speed in guar syrup, which is inconsistent with the researchers' conclusion.