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
case1	8	4	5	3
case2	7	2	4	4
case3	8	3	4	2
case4	6	4	3	3

a. What are the independent variable(s) and dependent variable of this study?

b. Write down the null hypothesis and the alternative hypothesis for each of the effects in the analysis.

c. 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

a. What are the independent variable(s) and the dependent variable of this study?

Ans:  The independent variable is relationship status (single vs. in a relationship) whereas the dependent variable is the perceived level of loneliness of the study.

b. Write down the null hypothesis and the alternative hypothesis for each of the effects in the analysis.

Ans: Null Hypothesis: The mean level of loneliness for all 4 groups of relationship status are the same.

Alternative Hypothesis: At least one group of relationship status has a significant difference mean of the level of loneliness from the other 3 groups.

c. 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

ANOVA table

Source	DF	SS	MS	F	P
Between	3	46.25	15.417	19.47	0.000
Within	12	9.500	0.792
Total	15	55.75

S = 0.8898 R-Sq = 82.96% R-Sq(adj) = 78.70%

Decision: The estimated p-value is less than 005 level of significance. Hence, reject the null hypothesis.

Conclusion: 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 at the 005 level of significance.