Question

A study of the pay of corporate chief executive officers (CEOs) examined the increase in cash compensation of the CEOs of 103 companies, adjusted for inflation, in a recent year. The mean increase in real compensation was x = 7.7%, and the standard deviation of the increases was s = 41%. Is this good evidence that the mean real compensation μ of all CEOs increased that year?

Ho:	μ = 0	(no increase)
Ha:	μ > 0	(an increase)

Because the sample size is large, the sample s is close to the population σ, so take σ = 41%.

(a) Sketch the normal curve for the sampling distribution of x when H_o is true. Shade the area that represents the P-value for the observed outcome x = 7.7%. (Do this on paper. Your instructor may ask you to turn in this work.)

(b) Calculate the P-value. (Round your answer to four decimal places.)


(c) Is the result significant at the α = 0.05 level? Do you think the study gives strong evidence that the mean compensation of all CEOs went up?

Reject the null hypothesis, there is significant evidence that the mean compensation of all CEOs went up.

Reject the null hypothesis, there is not significant evidence that the mean compensation of all CEOs went up.    

Fail to reject the null hypothesis, there is not significant evidence that the mean compensation of all CEOs went up.

Fail to reject the null hypothesis, there is significant evidence that the mean compensation of all CEOs went up.

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u = 0
Alternative hypothesis: u > 0

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a one-sample t-test.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = s / sqrt(n)

S.E = 4.0399
DF = n - 1

D.F = 102
t = (x - u) / SE

t = 1.906

where s is the standard deviation of the sample, x is the sample mean, u is the hypothesized population mean, and n is the sample size.

a)

The observed sample mean produced a t statistic test statistic of 1.906.

b)

Thus the P-value in this analysis is 0.0297.

c) Interpret results. Since the P-value (0.0297) is less than the significance level (0.05), hence we have to reject the null hypothesis.

Reject the null hypothesis, there is significant evidence that the mean compensation of all CEOs went up.