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	8	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.

Answer 1

H0: Null Hypothesis: ( People’s level of loneliness does not 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)

HA: Alternative Hypothesis: (at least one mean is different from other 3 means) ( People’s level of loneliness 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)

From the given data, the following Table is calculated:

	N	Mean	Std. Dev.
Case 1	4	7.75	0.5
Case 2	4	3.25	0.9574
Case 3	4	4.00	0.8165
Case 4	4	3.00	0.8165

From the above Table, ANOVA Table is calculated as follows:

Source of Variation	Degrees of freedom	Sum of squares	Mean squares	F - Stat	P - Value
Between Groups	3	58.5	58.5/3=19.5	19.5/0.625=31.2005	0.0000
Within Groups	12	7.4999	7.4999/12=0.625
Total	15	65.9999

F - Stat = 19.5/0.625=31.2005

Degrees of freedom for numerator = 3

Degrees of freedom for denominator = 12

By Technology, P - value = 0.0000

Since P - Value is less than = 0.05, the difference is significant. Reject null hypothesis.

Conclusion:
The data support the claim that people’s level of loneliness 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