Question

Contingency tables may be used to present data representing scales of measurement higher than the nominal scale. For example, a random sample of size 20 was selected from the graduate students who are U.S. citizens, and their grade point averages were recorded. 3.42 3.54 3.21 3.63 3.22 3.8 3.7 3.2 3.75 3.31 3.86 4 2.86 2.92 3.59 2.91 3.77 2.7 3.06 3.3 Also, a random sample of 20 students was selected from the non-U.S. citizen group of graduate students at the same university. Their grade point averages were as follows. 3.50 4.00 3.43 3.85 3.84 3.21 3.58 3.94 3.48 3.76 3.87 2.93 4.00 3.37 3.72 4.00 3.06 3.92 3.72 3.91 Test the null hypothesis that the proportion of graduate students with averages of 3.50 or higher is the same for both the U.S. citizens and the non-U.S. citizens

Answer 1

We have given

Let p1: proportion of graduate students with averages of 3.50 or higher for both the U.S. citizens.

p2: proportion of graduate students with averages of 3.50 or higher for both the non U.S. citizens.

The value of the pooled proportion is computed as

(1) Null and Alternative Hypotheses

This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is

The rejection region for this two-tailed test is

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that , it is then concluded that the null hypothesis is not rejected.

(5) Conclusion

There is not enough evidence to claim that the population proportion p1​ is different than p2​, at the 0.05 significance level.

i.e.  the proportion of graduate students with averages of 3.50 or higher is the same for both the U.S. citizens and the non-U.S. citizens