Question

Q.4 A researcher is interested in whether people’s level of loneliness would vary as a function of their relationship status (single vs. in a relationship), and how such difference might depend on whether people own a pet or not. She recruited a group of participants, asking them about their relationship status, pet ownership, and the perceived level of loneliness. The data are as below, with a higher number denoting greater level of loneliness:

	Single/ no pet	In a Relationship/ no pet	Single/ have pet	In a Relationship/ have pet
Case 1	8	4	5	3
Case 2	7	2	4	4
Case 3	8	3	4	2
Case 4	6	4	3	3

Conduct a proper statistical test by hand calculation to test the hypotheses in b., with 5% as the level of significance (α). (For this exercise, the data assumptions of your chosen statistical test can be taken as reasonably met.) Show your calculation formulae and steps. In case you decide to conduct an ANOVA, you are not required to conduct any post-hoc comparisons. Decide whether to reject the null hypothesis or not for each effect and state the basis of your decision.

*Sorry, reuploaded the question because I wrote the wrong numbers

Answer 1

This is One-Way ANOVA analysis because we have two variables, namely, relationship status and whether people own a pet or not.

The hypothesis being tested is:

H₀: µ₁ = µ₂ = µ₃ = µ₄

H_a: At least one means is not equal for all groups

The output is:

	Mean	n	Std. Dev
	7.3	4	0.96	Group 1
	3.3	4	0.96	Group 2
	4.0	4	0.82	Group 3
	3.0	4	0.82	Group 4
	4.4	16	1.93	Total
ANOVA table
Source	SS	df	MS	F	p-value
Treatment	46.25	3	15.417	19.47	.0001
Error	9.50	12	0.792
Total	55.75	15

Since the p-value (0.0001) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that at least one group of relationship status has a significant difference mean of the level of loneliness from the other 3 groups.