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

Click here for the Excel Data File

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.

• 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.)

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

Applying one way ANOVA: (use excel: data: data analysis: one way ANOVA: select Array):

Source	SS	df	MS	F	P value	Fcrit
Between	146.80	2	73.400	4.926	0.027	3.885
Within	178.80	12	14.9000
Total	325.60	14

a)

  grand mean =23.60

b)

SSTR=146.80

MSTR =73.40

c)

SSE =178.80

MSE =14.90

d)

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

e)

0.025 p-value < 0.05

f)

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

g)

We conclude that some means differ.