Question

A researcher believes that the mean earnings of top-paid actors, thletes and musicians are the same. The earnings (in millions of dollars) for several randomly selected people from each category are shown in the table below. It has been confirmed that the population variance for each group is equal and that earnings follow a normal distribution for top-pain actors, athletes and musicians. Use a 5% significance to test the claim.

Actor	Athlete	Musician
31	45	43
30	41	52
33	50	53
24	21	67
30	51	36
25	41	63
22	43	39
24	47	28
36	31	51
28	34	43
28		26
30		66
27		39
31		43
		39

Is the factor variable top-paid job type or earnings?
Is the response variable top paid job type or earnings?
Test Statistic:
P-Value:
Do we reject or fail to reject the null hypothesis?
Did Something Significant Happen?
There is or is not to conclude that they get paid the same or they don't?

Answer 1

Null hypothesis is that mean earnings of top-paid actors, athletes and musicians are the same.

The factor variables are top-paid job type.

The response variable is earnings.

Since population variance for each group is equal and that earnings follow a normal distribution for top-pain actors, athletes and musicians, all assumptions of one way ANOVA test are satisfied. We will conduct one way ANOVA test for the hypothesis.

Level of significance = 0.05

Degree of freedom of group = Number of level - 1 = 3 - 1 = 2

Degree of freedom of error = Number of observations - Number of level = 39 - 3 = 36

Critical value of F at DF = 2,36 is 3.26

Let T_i be the total earnings for group i, n_i be number of observations of group i.

Let G be the total earnings of all observations and N be total number of observations.

ΣX² is sum of squares of all observations (earnings)

T1 = 399, T2 = 404 , T3 = 688

G = 399 + 404 + 688 = 1491

ΣX² = 11565 + 17104 + 33794 = 62463

SST = ΣX² - G²/N = 62463 - 1491²/39 = 5460.923

SSTR = ΣT²/n - G²/N = (399² /14 + 404² /10 + 688² /15 ) - 1491²/39 = 2247.29

SSE = 5460.923 - 2247.29 = 3213.633

MSTR = SSTR / DF Group = 2247.29 / 2 = 1123.645

MSE = SSE / DF Error = 3213.633 / 36 = 89.26758

F Test Statistic = MSTR / MSE = 1123.645 / 89.26758 = 12.587

P-value = P(F > 12.587, df = 2, 36) = 0.00007

As, p-value is less than the significance level of 0.05 (or the observed value of F (12.587) is greater than the critical value (3.26), we reject the null hypothesis and conclude that the at least one top-paid job type have significant different earnings than other job types.

Yes, we reject the null hypothesis.

Yes, at least one top-paid job type have significant different earnings than other job types.

There is not to conclude that they get paid the same.