Question

A manager wishes to see if the time (in minutes) it takes for their workers to complete a certain task is different if they are wearing earbuds.   A random sample of 20 workers' times were collected before and after wearing earbuds. Test the claim that the time to complete the task will be different, i.e. meaning has production differed at all, at a significance level of α = 0.01

For the context of this problem, μ_D = μ_before−μ_after where the first data set represents before earbuds and the second data set represents the after earbuds. Assume the population is normally distributed. The hypotheses are:

H₀: μ_D = 0
H₁: μ_D  0

Before

After

69

65.3

69.5

61.6

39.3

21.4

66.7

60.4

38.3

46.9

85.9

76.6

70.3

77.1

59.8

51.3

72.1

69

79

83

61.7

58.8

55.9

44.7

56.8

50.6

71

63.4

80.6

68.9

59.8

35.5

73.1

77

49.9

38.4

56.2

55.4

64.3

55.6

Answer 1

Before	After	Difference
69	61.7	7.3
65.3	58.8	6.5
69.5	55.9	13.6
61.6	44.7	16.9
39.3	56.8	-17.5
21.4	50.6	-29.2
66.7	71	-4.3
60.4	63.4	-3
38.3	80.6	-42.3
46.9	68.9	-22
85.9	59.8	26.1
76.6	35.5	41.1
70.3	73.1	-2.8
77.1	77	0.1
59.8	49.9	9.9
51.3	38.4	12.9
72.1	56.2	15.9
69	55.4	13.6
79	64.3	14.7
83	55.6	27.4
	Average	4.245
	Stdev.	20.148

The following null and alternative hypotheses need to be tested:

Ho: μD​ = 0

Ha: μD​ ≠ 0

This corresponds to a two-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=.01, and the degrees of freedom are df = 19.

Hence, it is found that the critical value for this two-tailed test is tc​=2.861, for α=.01 and df = 19.

The rejection region for this two-tailed test is R={t:∣t∣>2.861}.

(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 , it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.3579, and since , it is concluded that the null hypothesis is not rejected.

Thus there is not enough evidence to say that the time differs if the workers wear an earbud.