Question

12-E2.  A survey of 1547 customers involved several questions that required yes/no answers. The surveys were administered by 6 different interviewers. The results below relate to the question "I think that luxury cars are not worth the money."

Interviewer	1	2	3	4	5	6
Agree	38	60	14	102	68	65
Disagree	127	221	102	323	202	225

1. How much evidence is there that interviewer 3 is different from the other5 treated collectively?
2. Would it be reasonable to combine the data into a single result, namely 347 agree out of 1547 surveyed?

Answer 1

a) Proportion of customers for Interviewer 3 who agreed, p1 = 14 / (14+102) = 0.121

Proportion of customers for other interviewers who agreed,

p2 = (38+60+102+68+65) / (38+60+102+68+65+127+221+323+202+225)

= 0.233

We want to test if the proportions p1 and p2 are statistically different. Hence, we do the 2 samples proportions difference test.

Difference between the samples, p' = 0.121 - 0.233 = -0.212

Hypothesized difference between the samples = 0

Now, we calculate the Pooled proportion, Pa = (p1*n1+p2*n2)/(n1+n2) = 0.222

Z-statistic = p' / sqrt[ Pa(1-Pa)*(1/n1 + 1/n2) ] = 1.156

Now, Z-critical for significance level of 5%, Zc = 1.645

Since Z-statistic < 1.645, we cannot reject the null hypothesis at the 5% significance level. Hence, the 2 samples means are statistically not different at the 5% significance level.

b) As seen above, since the proportions are statistically similar at the 5% significance level, it might be reasonable to combine the data into a single result.