Question

Suppose you wanted to examine whether men and women differ with regard to how many names they tend to mention. (For convenience, we will refer to those named as “close friends.”)

Number of Close Friends    0 1 2 3 4 5 6 total

Number of Respondents (male) 196 135 108 100 42 40 33 654

Number of Respondents (women) 201 146 155 132 86 56 37 813

m. Conduct a test of whether men and women differ with regard to the proportion who respond with zero names to the survey question asking who they talk to about important matters. Report all aspects of the test (all five steps as shown in your text), and summarize your conclusion.

Answer 1

= The sample proportion of men who responded to having 0 close friends = 196 / 654 = 0.2997

= The sample proportion of women who responded to having 0 close friends = 201 / 813 = 0.2472

Let = Overall proportion = (196 + 201)/ (654 + 813) = 397 / 1467 = 0.2706

1 - = 0.7294

= Default level (as nothing is mentioned) = 0.05

(a) The Hypothesis:

H0: p₁ = p₂ : The proportion of men who responded to having 0 close friends is equal to the proportion of women who responded to having 0 close friends.

Ha: p₁ p₂ : The proportion of men who responded to having 0 close friends is different from the proportion of women who responded to having 0 close friends.

This is a Two Tailed Test.

The Test Statistic:

The p Value: The p value (Two Tail) for Z = 2.25, is; p value = 0.0244

The Critical Value: The critical values (Two tail) at = 0.05, Zcritical = + 1.96 and -1.96

The Decision Rule:    

Critical Value Method: If Z observed is < - Zcrituical or if Zobserved is > Zcritical, then Reject H0.

P-Value Method: If the P value is < , Then Reject H0

The Decision:   

Critical Value Method: Since Z observed (2.25) is > Zcritical (1.96), We Reject H0.

P-Value Method: Since P value (0.0244) is < (0.05), We Reject H0.

The Conclusion: There is sufficient evidence at the 95% significance level to conclude that the proportion of men who responded to having 0 close friends is different from the proportion of women who responded to having 0 close friends.