Question

Researchers gave a family planning survey was given to the head(s) of household in a local community. In particular, one question asked the head(s) of households to indicate if they think the community has a great need, some need, or no need of family planning counseling. In addition, they were asked to indicate how many children are in the household. Researchers want to know if the number of children in the household is impacted by family planning counseling. What can they conclude with α = 0.01?

great need	some need	no need
0 0 2 3 1 2 0 1	6 2 3 1 5 4 3 5	9 4 3 2 9 4 5 4

Compute the appropriate test statistic(s) to make a decision about H₀.
critical value =

Conduct Scheffe's Post Hoc Test for the following comparisons:
1 vs. 2: test statistic =   ;   
1 vs. 3: test statistic =   ;   

Answer 1

Treatment →	great need	some need	no need	Pooled Total
observations N	8	8	8	24
sum ∑xi	9	29	40	78
mean ¯x	1.125	3.625	5	3.25
sum of squares ∑xi2	19	125	248	392

Hypothesis:

H₀ : All means are equal

H₁: Atleast one mean is not equal.

ANOVA Table

Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	61.75	2	30.875	8.447883	0.002033	5.780
Within Groups	76.75	21	3.654762
Total	138.5	23

Since the test statistic F (8.45) is greater than critical value 5.780, hence we reject null hypothesis and there is a significant evidence to conclude that atleast one treatment mean is different.

Tukey HSD results

treatments pair	Tukey HSD Q statistic	Tukey HSD p-value	Tukey HSD inferfence
1 vs. 2:	3.6988	0.041024	* p<0.05
1 vs. 3:	5.7331	0.001586	** p<0.01
2 vs. 3:	2.0343	0.340529	insignificant

Conduct Scheffe's Post Hoc Test for comparisons:

treatments pair	Scheffé test statistic	Scheffé p-value	Scheffé inferfence
1 vs. 2:	2.6154	0.051786	insignificant
1 vs. 3:	4.0539	0.002312	** p<0.01
2vs. 3:	1.4385	0.372784	insignificant