Question

Question number 12

A manager wishes to see if the time (in minutes) it takes for their workers to complete a certain task is faster if they are wearing ear buds.   A random sample of 20 workers' times were collected before and after wearing ear buds. Test the claim that the time to complete the task will be faster, i.e. meaning has production increased, 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 ear buds and the second data set represents the after ear buds. Assume the population is normally distributed. The hypotheses are:

H₀: μ_D = 0
H₁: μ_D > 0

You obtain the following sample data:

Before	After
69	62.3
71.5	61.6
39.3	21.4
67.7	60.4
38.3	47.9
85.9	77.6
67.3	75.1
59.8	46.3
72.1	65
79	83
61.7	56.8
55.9	44.7
56.8	50.6
71	63.4
80.6	68.9
59.8	35.5
72.1	77
49.9	38.4
56.2	55.4
63.3	51.6

a) Find the p-value. Round answer to 4 decimal places.

Answer:

b) Choose the correct decision and summary.

Do not reject H0, there is not enough evidence to support the claim that the time to complete the task has decreased when workers are allowed to wear ear buds at work.

Do not reject H0, there is enough evidence to support the claim that the time to complete the task has decreased when workers are allowed to wear ear buds at work.

Reject H0, there is enough evidence to support the claim that the time to complete the task has decreased when workers are allowed to wear ear buds at work.

Reject H0, there is not enough evidence to support the claim that the time to complete the task has decreased when workers are allowed to wear ear buds at work.

Answer 1

= (6.7 + 9.9 + 17.9 + 7.3 + (-9.6) + 8.3 + (-7.8) + 13.5 + 7.1 + (-4) + 4.9 + 11.2 + 6.2 + 7.6 + 11.7 + 24.3 + (-4.9) + 11.5 + 0.8 + 11.7)/20 = 6.715

s_d = sqrt(((6.7 - 6.715)^2 + (9.9 - 6.715)^2 + (17.9 - 6.715)^2 + (7.3 - 6.715)^2 + (-9.6 - 6.715)^2 + (8.3 - 6.715)^2 + (-7.8 - 6.715)^2 + (13.5 - 6.715)^2 + (7.1 - 6.715)^2 + (-4 - 6.715)^2 + (4.9 - 6.715)^2 + (11.2 - 6.715)^2 + (6.2 - 6.715)^2 + (7.6 - 6.715)^2 + (11.7 - 6.715)^2 + (24.3 - 6.715)^2 + (-4.9 - 6.715)^2 + (11.5 - 6.715)^2 + (0.8 - 6.715)^2 + (11.7 - 6.715)^2)/19) = 8.44

a) The test statistic t = ( - D)/(s_d/)

                                = (6.715 - 0)/(8.44/)

                                = 3.56

P-value = P(T > 3.56)

           = 1 - P(T < 3.56)

           = 1 - 0.9990

          = 0.0010

Since the P-value is less the significance level(0.001 < 0.01), so we should reject the null hypothesis.

b) Reject H0, there is enough evidence to support the claim that the time to complete the task has decreased when workers are allowed to wear ear buds at work.