STAT460 - Experimental Design
Daily Homework #6 - Testing Proportions

Review questions:

1) What is a hypothesis test? How is one conducted?

2) What is meant by a statistically significant result?

3) What is the z-test for one proportion? What assumptions are necessary for it to be valid?

4) What is the continuity correction? When is it used, and why?

5) What is the rejection region of a hypothesis test?

Computational exercises:

1*) In a "bold gamble to revive depressed sales," the Joseph Schlitz Brewing Co. announced that it would broadcast on live television a taste test featuring 100 beer drinkers during half time of a National Football League playoff game. During the live broadcast, Schlitz claimed that the 100 beer drinkers selected for the taste test were "loyal" drinkers of Budweiser, the industry's best-selling beer. Each of the participants was served two beers, one Schlitz and one Budweiser, in unlabelled ceramic mugs. Tasters were then told to make a choice by pulling an electronic switch left or right in the direction of the beer they preferred. (Prior to the test, the tasters were informed that one of the mugs contained their regular beer, Budweiser, and the other contained Schlitz, but the ordering was not revealed.) The percentage of the 100 "loyal" Budweiser drinkers who preferred Schlitz was then tabulated live, in front of millions of football fans. The results of the live TV taste test showed that 46 of the 100 "loyal" Budweiser drinkers preferred Schlitz. Schlitz, of course, labeled the outcome "an impressive showing" in a magazine advertisement following the test. Dr. Rasp is a bit more cynical about the whole process. He claims that most people can't tell one beer from another (or from another liquid of similar color, for that matter). Are the data consistent with Dr. Rasp's explanation?

2) Willard H. Longcor** conducted some tests of inexpensive and precision-made dice. Two million rolls were made with the precision-made dice, recording on each roll whether an even or odd number appeared. A new die was used after every 20,000 tosses, to guard against imperfections from the wear and tear of being rolled over and over. The same experiment was conducted with inexpensive dice, but Longcor stopped after 1,160,000 rolls. Results from his experiment are given below:

a) For each type of dice, test the null hypothesis that even and odd numbers are equally likely to be rolled.

b) What is the rejection region, for the test with n=2,000,000?

* I stole (er, "liberated") this problem from Gary's Smith's textbook Statistical Reasoning.

** I'm NOT making this name up. The experiment is reported in Mosteller, Rourke, and Thomas's book Probability with Statistical Applications.

SOLUTIONS:
1) z=0.90, using continuity correction.
   p-value .1941 [one-tailed].
   Don't reject H0.
2) precision dice z=1.27, p-value=.2040, don't reject null hypothesis
   inexpensive dice z=15.6, reject null hypothesis
   rejection region: p<.499307 and p>.500693