Question

A researcher claims that a​ post-lunch nap decreases the amount of time it takes males to sprint 20 meters after a night with only 4 hours of sleep. The table shows the amounts of time​ (in seconds) it took for 10 males to sprint 20 meters after a night with only 4 hours of sleep when they did not take a​ post-lunch nap and when they did take a​ post-lunch nap. At

alphaαequals=0.050.05​,

is there enough evidence to support the​ researcher's claim? Assume the samples are random and​ dependent, and the population is normally distributed. Complete parts​ (a) through​ (e) below.

Male

1

2

3

4

5

6

7

8

9

10

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Sprint time​ (without nap)

3.933.93

3.983.98

4.024.02

4.014.01

4.044.04

3.993.99

3.953.95

3.913.91

3.973.97

3.993.99

Sprint time​ (with nap)

3.923.92

3.983.98

4.004.00

4.034.03

4.014.01

3.983.98

3.963.96

3.883.88

3.963.96

3.963.96

Answer 1

The following table is obtained:

	Sample 1	Sample 2	Difference = Sample 1 - Sample 2
	3.93	3.92	0.01
	3.98	3.98	0
	4.02	4.00	0.02
	4.01	4.03	-0.02
	4.04	4.01	0.03
	3.99	3.98	0.01
	3.95	3.96	-0.01
	3.91	3.88	0.03
	3.97	3.96	0.01
	3.99	3.96	0.03
Average	3.979	3.968	0.011
St. Dev.	0.04	0.044	0.017
n	10	10	10

For the score differences, we have

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

This corresponds to a right-tailed test, for which a t-test for two paired samples be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=9.

Hence, it is found that the critical value for this right-tailed test is tc​=1.833, for α=0.05 and df = 9

The rejection region for this right-tailed test is R=t:t>1.833.

(3) Test Statistics

The t-statistic is computed as shown in the following formula:

(4) Decision about the null hypothesis

Since it is observed that t=2.012>tc​=1.833, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0375, and since p=0.0375<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ1​ is greater than μ2​, at the 0.05 significance level.

Graphically

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