These practice problems emphasize various aspects of
hypothesis testing. Some of them are a bit "non traditional", because
we hope to emphasize the general framework behind hypothesis testing,
rather than provide practice on particular tests.
1) Go to the UCLA case studies link and examine
whether Dr. Spock (the pediatrician, not the fictional vulcan)
received a fair trial. If you click on the "explain" link at the bottom
of the page, you'll see a solution.
2) We don't want your students to do calculate power probabilities. But
you should do it once or twice to get the hang of it. So here are two
problems to give you a chance. Suppose I have a handful of "trick"
dice. These are dice that have been altered to produce odd numbers more
frequently than even numbers. I hand you a die, and you determine if
it's a trick die or a fair die. And let's agree, for now, to make this
determination the following way: roll the die 10 times and let X
represent the number of times an odd number appears. If X > 8, we'll
reject the null hypothesis (the die is fair) and conclude the die is
not fair. One important lesson: there is not one power. The value of
the power depends on the value of the population parameter, and hence
there is a different probability for each value of the "true" parameter.
a) Suppose we do this experiment with a fair die. What's the
probability that we'll reject the null hypothesis. (Note, this
probability is the significance level: P(reject H0 given that H0 is
true).
b) The next die I show you has been altered so that the probability of
it landing on an odd number is .60. Now what's the probability that
we'll reject the null hypothesis? This probability is the power when
p=.6 : P(reject H0 given that it's false and, more specifically, equal
to .6).
c) The next die I show you has been altered so that the probability of
it landing on an odd number is .75. Now what's the probability that
we'll reject the null hypothesis? Notice that it is larger than for the
other die. The further the "true" value gets from what the null
hypothesis says, the better your chances of picking this up in the
significance test.
d) Power is also affected by the significance level. In part (a) you
calculated the significance level, which was determined by our decision
to put the "cutoff" point at 9. This means that we determined to
reject the null hypothesis if the number of rolls that resulted in odd
was 9 or 10. Let's change the significance level by changing the
cutoff. Recalculate the significance level if we used 8 as a cutoff
rather than 9. What affect does this have on the power you calculated
in (b) and (c)?
3) Let's try a slightly more realistic calculation. When I go to the
campus food court, I order a 16 ounce diet coke with my meal. They have
a fancy machine that's supposed to dispense exactly 16 ounces. Let's
say that I have my own fancy machine that measures how many ounces they
actually gave me. Because of the froth and foam, I don't expect to get
exactly 16 ounces every time. In fact, because I looked it up on the
web site of the manufacturers of the machine, the standard deviation
for the amount of soda dispensed is 0.2 ounces. But still, if the mean
is 16 ounces, then my average amount should be 16. Being the penny
pincher that I am, I will lodge loud and vociferous complaints should I
suspect that the mean is actually less than 16, and I shall remain
quiet and smug if the mean is bigger than 16. Suppose, for the sake of
argument, that after 10 sodas, I will calculate the average of these 10
sodas and decide whether to lodge a complaint or remain silent.
a) Write out the null and alternative hypotheses.
b) Suppose we want to use a 5% significance level. We will calculate
the average of our 10 sodas and reject the null hypothesis if the
average is less than or equal to what number?
c) Suppose that, unbeknownst to me, the greedy SOBs at Fast Food
Friends have set their soda machine to put in 15.8 ounces of soda.
What's the probability that my method will detect this fiendish plot?
4) OK, now for a matteroffact one. From Devore and Peck. The
recommended daily dietary allowance for zinc among males older than 50
years is 15 mg/day. A group of medical researchers was concerned that
men aged 6574 years old in the U.S. were not getting sufficient levels
of zinc in their diet. Based on a sample of 115 men in this age group,
the average zinc intake was 11.3 mg/day nd the standard deviation of
these 115 men was 6.43. Do a hypothesis test to determine whether the
mean zinc intake for all men in this age group fails to exceed the
recommended daily allowance. Use a 5% significance level.
Solutions to 24.
