A random sample of five observations from three normally distributed populations produced the following data: (You may find it useful to reference the F table.)
Treatments | ||||||||||
A | B | C | ||||||||
15 | 22 | 17 | ||||||||
23 | 28 | 29 | ||||||||
31 | 18 | 24 | ||||||||
32 | 15 | 23 | ||||||||
23 | 25 | 27 | ||||||||
x−Ax−A |
= |
24.8 | x−Bx−B | = | 21.6 | x−Cx−C | = | 24.0 | ||
s2AsA2 | = | 48.2 | s2BsB2 | = | 27.3 | s2CsC2 | = | 21.0 | ||
a. Calculate the grand mean. (Round intermediate calculations to at least 4 decimal places and final answer to 4 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 1% 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 αα = 0.01.
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.
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