Question

Suppose a simple random sample of 26 male students is asked whether they have pulled an "all-nighter " for academic reasons and 10 answered "yes." Suppose a simple random sample of 25 female students is asked whether they have pulled an "all-nighter" for academic reasons and 8 answered "yes."

Is there insufficient evidence based on the sample to reject the claim that male and female students are equally likely to have pulled an "all-nighter" for academic reasons?

Answer 1

Let be the true proportion of male students who have pulled an "all-nighter" for academic reasons and

be the true proportion of male students who have pulled an "all-nighter" for academic reasons.

We want to the claim that male and female students are equally likely to have pulled an "all-nighter" for academic reasons, that is we want to test if

The hypotheses are

We have the following information from the samples

An estimate of overall proportion of students who have pulled an "all-nighter" for academic reasons is

The estimated standard error of mean difference in proportions is

The hypothesized value of the mean difference is

The tets statistic is

We will test the hypotheses at 5% level of significance.

This is a 2 tailed test (the alternative hypothesis is "not equal to").

The right tail critical value is

Using the standard normal tables, we get for z=1.96, P(Z<1.96)=0.975

The critical values are -1.96 and +1.96

We will reject the null hypothesis, if the tets statistic does not lie in the acceptance region -1.96 to +1.96

Here, the test statistic is 0.4827 and it lies within the interval -1.96 and +1.96. Hence we do not reject the null hypothesis.

Ans: We conclude that there is insufficient evidence based on the sample to reject the claim that male and female students are equally likely to have pulled an "all-nighter" for academic reasons.