Question

Section 1	Section 2	Section 3
82.5	68.1	71.1
73.9	83.8	77.4
78.8	71.9	78.2
71.9	45.1	84.8
84.5	69.1	82.3
70.8	61.8	78.8
65.8	87.4	75.6
88.2	47.7	74.3
79.7	60.9	69.1

Run a one-factor ANOVA (fixed effect) with α=0.05 Report the F-ratio to 3 decimal places and the P-value to 4 decimal places.
F=3.787976
p=0.0371755
What is the conclusion from the ANOVA?

• reject the null hypothesis: at least one of the group means is different
• fail to reject the null hypothesis: not enough evidence to suggest the group means are different

Calculate the group means for each section:
Section 1:  M1=77.34
Section 2:  M2=66.200
Section 3:  M3=76.84
Report means accurate to 2 decimal places.
NEED HELP WITH THIS NEXT SECTION

Conduct 3 independent sample t-tests (2-tail) for each possible pair of sections. (Though we will see later that it might not be appropriate, retain the significance level α=0.05α=0.05.) Report the P-value (accurate to 4 decimal places) for each pairwise comparison.
Compare sections 1 & 2:  p=
Compare sections 1 & 3:  p=
Compare sections 2 & 3:  p=

Answer 1

The one-way analysis of variance (ANOVA) is used to determine whether there are any statistically significant differences between the means of three or more independent groups. In this example

Specifically, it tests the null hypothesis:

Ho:μ1=μ2=μ3

where = i^th group mean.

Alternative hypothesis is there are at least two group means that are statistically significantly different from each other.

After Running a one-factor ANOVA (fixed effect) with α=0.05 the summary is as follow.

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
Section 1	9	696.1	77.34444	52.61278
Section 2	9	595.8	66.2	204.8275
Section 3	9	691.6	76.84444	25.00278
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	713.2585	2	356.6293	3.787977	0.037176	3.402826
Within Groups	2259.544	24	94.14769
Total	2972.803	26

Conlusion:

If P-value is less than level of significance(α) we reject the null hypothesis.

Here P-value=0.037176 which is less than α=0.05 we reject the null hypothesis. that is there are at least two group means that are statistically significantly different from each other.

We need here further analyses that which group means are significantly different from each other. there are 3 possible pairs which are (Section 1, Section 2), (Section 1 Section 3), and (Section 2, Section3). WE Conduct t-tests for 3 independent pairs separately.

1) for pair (Section 1, Section 2). P-value(two tail)= 0.05923.

2) for pair (Section 1, Section 3). P-value(two tail)= 0.86724

3) for pair (Section 2, Section 3). P-value(two tail)= 0.0614