Question

# An executive at the home office of Big Rock Life Insurance is considering three branch managers...

An executive at the home office of Big Rock Life Insurance is considering three branch managers as candidates for promotion to vice president. The branch reports include records showing sales volume for each salesperson in the branch (in hundreds of thousands of dollars). A random sample of these records was selected for salespersons in each branch. All three branches are located in cities in which per capita income is the same. The executive wishes to compare these samples to see if there is a significant difference in performance of salespersons in the three different branches. If so, the information will be used to determine which of the managers to promote.

 Branch Managed by Adams Branch Managed by McDale Branch Managed by Vasquez 7.4 8.6 6.9 6.6 10.9 8.3 10.3 11.3 10.5 11.2 9.6 11.8 9.7 10.2

Use a 1% level of significance. Shall we reject or not reject the claim that there are no differences among the performances of the salespersons in the different branches?

(a) What is the level of significance?

State the null and alternate hypotheses.

Ho: μ1 = μ2 = μ3; H1: Not all the means are equal.Ho: μ1 = μ2 = μ3; H1: All three means are different.    Ho: μ1 = μ2 = μ3; H1: At least two means are equal.Ho: μ1 = μ2 = μ3; H1: Exactly two means are equal.

(b) Find

SSTOT, SSBET, and SSW

and check that

SSTOT = SSBET + SSW.

 SSTOT = SSBET = SSW =

Find

d.f.BET, d.f.W, MSBET, and MSW.

 d.f.BET = d.f.W = MSBET = MSW =

Find the value of the sample F statistic. (Round your answers to three decimal places.)

What are the degrees of freedom?
(numerator)
(denominator)

(c) Find the P-value of the sample test statistic.

P-value > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.001 < P-value < 0.010P-value < 0.001

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P value is greater than the level of significance at α = 0.01, we do not reject H0.Since the P value is less than or equal to the level of significance at α = 0.01, we reject H0.    Since the P value is greater than the level of significance at α = 0.01, we reject H0.Since the P value is less than or equal to the level of significance at α = 0.01, we do not reject H0.

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

At the 1% level of significance there is insufficient evidence to conclude that the means are not all equal.At the 1% level of significance there is sufficient evidence to conclude that the means are all equal.    At the 1% level of significance there is insufficient evidence to conclude that the means are all equal.At the 1% level of significance there is sufficient evidence to conclude that the means are not all equal.

 Source of Variation Sum of Squares Degrees of Freedom MS F Ratio P-Value Test Decision Between groups Within groups Total

a)

level of significance =0.01

Ho: μ1 = μ2 = μ3; H1: Not all the means are equal

b)

 SSTOT 37.384 SSBET 1.923 SSW 35.461
 dfBET 2 dfW 11 MSBET 0.961 MSW 3.224
 value of test statistic for factor A = 0.298
 df(numerator) = 2 df(Denominator) = 11

c)

P-value > 0.100

d)

Since the P value is greater than the level of significance at α = 0.01, we do not reject H0.

e_)

At the 1% level of significance there is insufficient evidence to conclude that the means are not all equal.

f_)

 Source SS df MS F P value decision Between 1.923 2 0.961 0.298 0.7480 fail to reject Ho Within 35.461 11 3.224 Total 37.384 13