Question

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
	25				17				22
	25				19				26
	27				25				26
	32				18				30
	18				17				27
x−Ax−A	=	25.4		x−Bx−B	=	19.2		x−Cx−C	=	26.2
s2AsA2	=	25.3		s2BsB2	=	11.2		s2CsC2	=	8.2

Treatments
A	B	C
25	17	22
25	19	26
27	25	26
32	18	30
18	17	27

a. Calculate the grand mean. (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)

Grand mean

b. Calculate SSTR and MSTR. (Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)

SSTR

MSTR

c. Calculate SSE and MSE. (Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)

SSE

MSE

d. Specify the competing hypotheses in order to determine whether some differences exist between the population means.

• H₀: μ_A = μ_B = μ_C; H_A: Not all population means are equal.

• H₀: μ_A ≤ μ_B ≤ μ_C; H_A: Not all population means are equal.

• H₀: 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.)

Test Statistic

e-2. Find the p-value.

• 0.025  p-value < 0.05
• 0.01  p-value < 0.025

• p-value < 0.01

• p-value  0.10
• 0.05  p-value < 0.10

f. At the 10% significance level, what is the conclusion to the test?

• Reject H₀ since the p-value is less than significance level.

• Do not reject H₀ since the p-value is not less than significance level.

• Do not reject H₀ since the p-value is less than significance level.

• Reject H₀ since the p-value is not less than significance level.

g. Interpret the results at αα = 0.10.

• We cannot conclude that some means differ.

• We conclude that some means differ.

• We conclude that all means differ.

• We conclude that population mean C is greater than population mean A.

Answer 1

using excel>data>data analysis>ANOVA

we have

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
A	5	127	25.4	25.3
B	5	96	19.2	11.2
C	5	131	26.2	8.2
		grand mean	23.6
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	146.8	2	73.4	4.926174	0.027422	3.885294
Within Groups	178.8	12	14.9
Total	325.6	14

a. Calculate the grand mean. (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)

Grand mean = 23.6000

b. Calculate SSTR and MSTR. (Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)

SSTR = 146.8

MSTR = 73.4

c. Calculate SSE and MSE. (Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.

SSE = 178.8

MSE = 14.9

d. Specify the competing hypotheses in order to determine whether some differences exist between the population means.

• H₀: 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.)

Test Statistic = 4.926

e-2. Find the p-value.

• 0.025 <  p-value < 0.05

f. At the 10% significance level, what is the conclusion to the test?

• Reject H₀ since the p-value is less than significance level.

g. Interpret the results at αα = 0.10.

• We conclude that some means differ.