Question

5) Will students wait times for the arrival of an instructor who is a full professor differ

from the wait times for an instructor who is a graduate student? This question was

investigated by counting how many minutes undergraduate students waiting in two

small seminar classes, one taught by a full professor and one taught by a graduate

student. Suppose we want to examine if the time waited is different for the full

professor than for the graduate student.

a. Propose a parametric approach to answer this question. Mention clearly all

assumptions for such an approach.

b. Compare your approach in part a. to a bootstrap approach for a two-sample

problem. Clearly state your conlusions in relation to this problem.

Answer 1

5.

(a)

We have to test equality of two population means where population variances are unknown. So, we have to perform two sample t-test.

For ease of our calculation an assumption about equality of population variances (though unknown) is considered. Further to calculate waiting time, we have to assume that both of them come sufficiently late than the scheduled time (and so students can wait sufficiently longer if desired.

Suppose, random variable X denotes waiting time in case of full time professor and random variable Y denotes that in case of an undergraduate student.

We have to test for null hypothesis

against the alternative hypothesis

So, in usual notations the test statistic is given by

where

We calculate the numerical value of test statistic as using numerical data obtained from sample.

Our degrees of freedom is

At certain level of significance , we calculate critical value as

We reject null hypothesis if and do not reject otherwise.

(b)

In bootstrap approach, we draw more samples with replacement from samples already obtained.

Suppose, we take m_x samples from sample obtained in case of full time professor and m_y samples in case of graduate student.

We then get m_x estimates for and m_y estimates for . We perform statistical inferences to obtain representative and as and .

Then we shall perform two sample t-test as already mentioned in part (a).