Question

In this problem, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more "conservative" answer.

Are America's top chief executive officers (CEOs) really worth all that money? One way to answer this question is to look at row B, the annual company percentage increase in revenue, versus row A, the CEO's annual percentage salary increase in that same company. Suppose a random sample of companies yielded the following data:

B: Percent increase for company	24	25	25	18	6	4	21	37
A: Percent increase for CEO	21	23	22	14	−4	19	15	30

Do these data indicate that the population mean percentage increase in corporate revenue (row B) is different from the population mean percentage increase in CEO salary? Use a 5% level of significance. (Let d = B − A.)

(A) What is the level of significance?

_______________


State the null and alternate hypotheses.

a) H₀: μ_d = 0; H₁: μ_d > 0

b) H₀: μ_d ≠ 0; H₁: μ_d = 0    

c) H₀: μ_d > 0; H₁: μ_d = 0

d) H₀: μ_d = 0; H₁: μ_d ≠ 0

e) H₀: μ_d = 0; H₁: μ_d < 0

(B) What sampling distribution will you use? What assumptions are you making?

a) The standard normal. We assume that d has an approximately uniform distribution.

b) The standard normal. We assume that d has an approximately normal distribution.    

c) The Student's t. We assume that d has an approximately uniform distribution.

d) The Student's t. We assume that d has an approximately normal distribution.

What is the value of the sample test statistic? (Round your answer to three decimal places.)

________________

(c) Find (or estimate) the P-value.

a) P-value > 0.500

b) 0.250 < P-value < 0.500    

c) 0.100 < P-value < 0.250

d) 0.050 < P-value < 0.100

e) 0.010 < P-value < 0.050

f) P-value < 0.010

Sketch the sampling distribution and show the area corresponding to the P-value.

Top right, top left, bottom right or bottom left?

(D) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

a) Since the P-value ≤ α, we reject H₀. The data are statistically significant.

b) Since the P-value > α, we fail to reject H₀. The data are not statistically significant.    

c) Since the P-value ≤ α, we fail to reject H₀. The data are statistically significant.

d) Since the P-value > α, we reject H₀. The data are not statistically significant.

(E) Interpret your conclusion in the context of the application.

a) Fail to reject H₀. At the 5% level of significance, the evidence is insufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary.

b) Reject H₀. At the 5% level of significance, the evidence is insufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary.    

c) Fail to reject H₀. At the 5% level of significance, the evidence is sufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary.

d) Reject H₀. At the 5% level of significance, the evidence is sufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary.

Answer 1

a)

0.05

d) H0: μd = 0; H1: μd ≠ 0

b)
d) The Student's t. We assume that d has an approximately normal distribution.

Test statistic,
t = (dbar - 0)/(s(d)/sqrt(n))
t = (2.5 - 0)/(7.5404/sqrt(8))
t = 0.938

c)

P-value Approach
P-value = 0.3795

b) 0.250 < P-value < 0.500

d)

b) Since the P-value > α, we fail to reject H0. The data are not statistically significant.

e)

a) Fail to reject H0. At the 5% level of significance, the evidence is insufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary.