In: Statistics and Probability
The following data are from a completely randomized design.
Treatment | |||
---|---|---|---|
A | B | C | |
163 | 142 | 125 | |
143 | 156 | 121 | |
165 | 124 | 138 | |
144 | 143 | 139 | |
147 | 136 | 149 | |
168 | 151 | 138 | |
Sample mean |
155 | 142 | 135 |
Sample variance |
132.4 | 127.6 | 105.2 |
(a)
Compute the sum of squares between treatments.
(b)
Compute the mean square between treatments.
(c)
Compute the sum of squares due to error.
(d)
Compute the mean square due to error. (Round your answer to two decimal places.)
(e)
Set up the ANOVA table for this problem. (Round your values for MSE and F to two decimal places, and your p-value to four decimal places.)
Source of Variation |
Sum of Squares |
Degrees of Freedom |
Mean Square |
F | p-value |
---|---|---|---|---|---|
Treatments | |||||
Error | |||||
Total |
(f)
At the α = 0.05 level of significance, test whether the means for the three treatments are equal.
State the null and alternative hypotheses.
H0: μA ≠
μB ≠ μC
Ha: μA =
μB =
μCH0:
μA = μB =
μC
Ha: μA ≠
μB ≠
μC H0:
Not all the population means are equal.
Ha: μA =
μB =
μCH0: At least two of the
population means are equal.
Ha: At least two of the population means are
different.H0: μA =
μB = μC
Ha: Not all the population means are equal.
Find the value of the test statistic. (Round your answer to two decimal places.)
Find the p-value. (Round your answer to four decimal places.)
p-value =
State your conclusion.
Reject H0. There is sufficient evidence to conclude that the means for the three treatments are not equal.Reject H0. There is not sufficient evidence to conclude that the means for the three treatments are not equal. Do not reject H0. There is not sufficient evidence to conclude that the means for the three treatments are not equal.Do not reject H0. There is sufficient evidence to conclude that the means for the three treatments are not equal.
The statistical software output for this problem is :
(a)
The sum of squares between treatments = 1236
(b)
The mean square between treatments = 618
(c)
The sum of squares due to error = 1826
(d)
The mean square due to error = 121.73
(e)
ANOVA table
Source | DF | SS | MS | F-Stat | P-value |
---|---|---|---|---|---|
Columns | 2 | 1236 | 618 | 5.08 | 0.0207 |
Error | 15 | 1826 | 121.733 | ||
Total | 17 | 3062 |
(f)
Ha: μA = μB = μC
Ha: At least two of the population means are different.
Test statistics = 5.08
P-value = 0.0207
Reject H0. There is sufficient evidence to conclude that the means for the three
treatments are not equal.