Are America s top chief executive officers CEOS really worth

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 that a random sample of companies yielded the following data: 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 1% level of significance. Are the data statistically significant at level alpha? Will you reject or fail to reject the null hypothesis?

Solution

let the population mean percentage increase in corporate revenue is mu1 and that of CEo salary is mu2

here we are to test H0: mu1=mu2 vs H1: mu1 is not equal to mu2

we have a random sample of size=n=8 from both the population

let X1bar be the sample mean of rowB and X2bar be that of row A

s1 be the sample SD of rowB and s2 be the sample SD of rowA

let s²=((n-1)*s1²+(n-1)*s2²)/(n+n-2)

consider the test statistic

T={(x1bar-x2bar)-(mu1-mu2)}/{s*sqrt(2/n)}

now under H0 mu1-mu2=0

so under H0 T={(x1bar-x2bar)}/{s*sqrt(2/n)}~t_n+n-2

we reject H0 iff |t|>t_{alpha/2;n+n-2} where t is the observed value of T and t_alpha;n+n-2 is the upper alpha point of a t distribution with df 2n-2

n=8 x1bar=15.63 x2bar=13.88 s1²=39.98 s2²=106.41 so s²=73.195   alpha=0.01 [as level of significance is 1%]

then observed value of t is 0.41

so p value=P[T>|0.41|]=0.689>0.01 where T~t_2n-2=14

hence the correct alternative is

since the interval containing the P values has values that are larger than the level of significance, the data are not statistically significant and so we fail to reject the null hypothesis.