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	26	25	23	18	6	4	21	37
A: Percent increase for CEO	21	23	20	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.

H₀: μ_d = 0; H₁: μ_d > 0H₀: μ_d > 0; H₁: μ_d = 0    H₀: μ_d = 0; H₁: μ_d < 0H₀: μ_d = 0; H₁: μ_d ≠ 0H₀: μ_d ≠ 0; H₁: μ_d = 0

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

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

P-value > 0.5000.250 < P-value < 0.500    0.100 < P-value < 0.2500.050 < P-value < 0.1000.010 < P-value < 0.050P-value < 0.010

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

(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?

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

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

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.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.    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.Fail to reject H₀. At the 5% level of significance, the e

Answer 1

a)The level of significance is given as α = 0.05

Null hypothesis:

H0: There is no difference exist in the mean percentage increase in corporate revenue (row B) is different from the population mean percentage increase in CEO salary, that is, μd = 0.

Alternative hypothesis:

H1: There is difference in the mean percentage increase in corporate revenue (row B) is different from the population mean percentage increase in CEO salary, that is, μd ≠0.

therefore,

H0: μd = 0; H1: μd ≠ 0. option 4 is correct.

b)Paired t-test statistic:

In order to test a hypothesis regarding whether the difference between a paired set of n observations (xi, yi) is significant or not, the paired t-test is used.

The student’s t, we assume that d has an approximately normal distribution.

The value of the sample statistic:

The test statistic for the paired t-test is given as follows:

t=dbar/(Sd/square root n)

patient	B: Percent increase for company	A: Percent increase for CEO	difference d=B-A
1	26	21	5
2	25	23	2
3	23	20	3
4	18	14	4
5	6	-4	10
6	4	19	-15
7	21	15	6
8	37	30	7
Total			22
Average			2.75

stndard deviation sd=7.592

t=2.75/(7.592/square root (8))

t=1.0245=1.025

df=8-1=7

c)p-value=2*p(t(1.025))

=2*0.339474

=0.67895=0.679

therefore,P-value > 0.5000.

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

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

e) 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.Fail to reject H0.