Random sampling from four normally distributed populations produced the following data: (You may find it useful to reference the F table.)
Treatments | ||||||||||||||
A | B | C | D | |||||||||||
−12 | −20 | −5 | −20 | |||||||||||
−20 | −9 | −18 | −20 | |||||||||||
−11 | −13 | −17 | −20 | |||||||||||
−19 | −7 | |||||||||||||
−16 | ||||||||||||||
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a. Calculate the grand mean. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)
b. Calculate SSTR and MSTR. (Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)
c. Calculate SSE and MSE. (Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)
d. Specify the competing hypotheses in order to determine whether some differences exist between the population means.
H_{0}: μ_{A} = μ_{B} = μ_{C}; H_{A}: Not all population means are equal.
H_{0}: μ_{A} ≤ μ_{B} ≤ μ_{C}; H_{A}: Not all population means are equal.
H_{0}: μ_{A} ≥ μ_{B} ≥ μ_{C}; H_{A}: Not all population means are equal.
e-1. Calculate the value of the F_{(df1, df2)} test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)
e-2. Find the p-value.
p-value < 0.01
f. At the 10% significance level, what is the conclusion to the test?
Do not reject H_{0} since the p-value is not less than significance level
Reject H_{0} since the p-value is less than significance level
Reject H_{0} since the p-value is not less than significance level
Do not reject H_{0} since the p-value is less than significance level
g. Interpret the results at α = 0.10.
We conclude that some means differ.
We cannot conclude that some means differ.
We cannot conclude that all means differ.
We cannot conclude that population mean C is greater than population mean A.
applying one way ANOVA
Source of Variation | SS | df | MS | F | P-value | F crit |
Between Groups | 105.1167 | 3 | 35.03889 | 1.383362 | 0.299223 | 3.587434 |
Within Groups | 278.6167 | 11 | 25.32879 | |||
Total | 383.7333 | 14 |
a) grand mean =-15.13
b)SSTR =105.1167
and MSTR =35.0389
c)
SSE =278.6167
MSE =25.3288
d)
H_{0}: μ_{A} = μ_{B} = μ_{C}; H_{A}: Not all population means are equal
e-1 )F =1.383
e-2)
p-value > 0.10
f)
Do not reject H_{0} since the p-value is not less than significance leve
g)
We cannot conclude that some means differ.
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