Question

You are curious about whether the professors and students at your school are of different political persuasions, so you take a sample of 20 professors and 20 students drawn randomly from each population. You find that 10 professors say they are conservative, and 6 students say they are conservative. Is this a statistically significant difference?

Please, I need a solution to this question. There are conflicting answers on the site; the one that seems reliable does not have an explanation on how the s.d. was calculated, and is one supposed to use the t-test or the z-test. Thanks

Answer 1

The sample proportions are first computed as:

p₁ = 10/20 = 0.5
p₂ = 6/20 = 0.3

Now we compute the pooled proportion as:
P = (10 + 6)/(20 + 20) = 16/40 = 0.4

The standard error for this Z test is computed here as:

Now the test statistic here is computed as:

As this is a two tailed test, the p-value here is computed from the standard normal table as:

p = 2P( Z > 1.29) = 2*0.099 = 0.1980

As the p-value here is 0.1980 which is higher than any general level of significance, therefore the test is not significant and we cannot reject the null hypothesis here. We dont have sufficient evidence to reject the null hypothesis that the two proportions are equal.