Question

For a hypothesis test of H0:p1 − p2 = 0 against the alternative Ha:p1 − p2 ≠ 0, the test statistic is found to be 3.31. Which of the following statements can you make about this finding?

The result is significant at both α = 0.05 and α = 0.01.

The result is significant at α = 0.05 but not at α = 0.01.

The result is significant at α = 0.01 but not at α = 0.05.

The result is not significant at either α = 0.05 or α = 0.01.

The result is inconclusive because we don't know the value of p.

Answer 1

Test Statistic = 3.31
Find the area under the standard normal Z curve, which is to the left of z = 3.31 to get this approximate result

P(Z < 3.31) = 0.9995

I used this Z table
http://users.stat.ufl.edu/~athienit/Tables/Ztable.pdf
scroll to the second page, then locate the row that starts with "3.3" and the column that has 0.01 at the top. The value 0.9995 is in this row and column combination

Since
P(Z < 3.31) = 0.9995
this means,
P(Z > 3.31) = 1-P(Z < 3.31)
P(Z > 3.31) = 1 - 0.9995
P(Z > 3.31) = 0.0005
The chances of getting a Z value that is beyond 3.31 is very small

Double this to get 2*0.0005 = 0.001
We double this area due to the fact that we have a two tailed hypothesis test

Therefore the p value is approximately 0.001
This p value is smaller than alpha = 0.05 and alpha = 0.01

So that's why the answer is
A) The result is significant at both a = 0.05 and a = 0.01
meaning we reject the null hypothesis at either alpha level and favor the alternative which says that \(\large p_1 - p_2 \ne 0\). This can be rewritten into \(\large p_1 \ne p_2\) (ie the population proportions of the two groups are likely to be different)