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.
H_{o}: μ_{1} = μ_{2} = μ_{3}; H_{1}: Not all the means are equal.H_{o}: μ_{1} = μ_{2} = μ_{3}; H_{1}: All three means are different. H_{o}: μ_{1} = μ_{2} = μ_{3}; H_{1}: At least two means are equal.H_{o}: μ_{1} = μ_{2} = μ_{3}; H_{1}: Exactly two means are equal.
(b) Find
SS_{TOT}, SS_{BET}, and SS_{W}
and check that
SS_{TOT} = SS_{BET} + SS_{W}.
(Round your answers to three decimal places.)
SS_{TOT} |
= | |
SS_{BET} |
= | |
SS_{W} |
= |
Find
d.f._{BET}, d.f._{W}, MS_{BET}, and MS_{W}.
(Round your answers to three decimal places.)
d.f._{BET} |
= | |
d.f._{W} |
= | |
MS_{BET} |
= | |
MS_{W} |
= |
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 H_{0}.Since the P value is less than or equal to the level of significance at α = 0.01, we reject H_{0}. Since the P value is greater than the level of significance at α = 0.01, we reject H_{0}.Since the P value is less than or equal to the level of significance at α = 0.01, we do not reject H_{0}.
(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.
(f) Make a summary table for your ANOVA test. (Round your answers
to three decimal places.)
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 |
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