MA 2611 E' 98 Test 1

1.
(25 points) In a study on the effect of caffeine intake on fine motor skills, 40 subjects were graded on their performance on a series of tests. The subjects were told to report at 8 am and were required to abstain from caffeine for 24 hours before that. The first 20 subjects to arrive were given a capsule of caffeine before performing the tests. The second 20 were given a capsule of sugar before performing the tests.
a.

ANS: A controlled experiment (10 points)

b.
There is an obvious flaw in the design of this study. Tell what it is, and how it can be corrected.

ANS: The subjects were not randomly assigned to treatment or control. The first 20 arrivals are likely to be early risers who don't need or react to caffeine differently than do the late arrivals. (15 points) 2

2.
(15 points) The half-life of a piece of uranium is the amount of time it takes for half the uranium atoms to decay. Suppose you have a data set consisting of the time to decay of each atom in a piece of uranium. What statistical measure of location for these times corresponds to the half-life of the uranium? Explain why.

ANS: The median, because the half-life is the time until half the uranium decays. In the data, it is the time value which half the lifetimes are greater than and half are smaller than: namely, the median of the lifetimes.

3.
(20 points) Zeolites are aluminum and silicon crystals which are used as sieves and catalysts in many industries-especially petroleum refining. Figure 1 shows frequency histograms of relative yields, in percent, of 30 samples taken from each of five consecutive production batches of zeolites.

a.
Is the process stationary? Why or why not?

ANS: No. The location of the histograms is dropping, from around 44 in batch 1 to around 39.5 in batch 5. (10 points)

b.
Suppose all five batches' worth of zeolite yields were plotted on a single histogram. Which of the histograms in Figure 2 would result?

ANS: Figure 2a would result. (10 points)

4.
(30 points) In a recent study, 200 runners were interviewed. It was found that those who stretched had substantially more injuries than runners who didn't.

a.

ANS: An observational study, since treatments (stretching) were not assigned to sampling units (runners). (10 points)

b.
Can we conclude from this that stretching causes injuries? Why or why not?

ANS: No, since this is not a controlled experiment, we can only conclude there is association between stretching and injuries. (10 points)

c.
If your answer to part b is no, what alternative explanation can you provide for the observed association?

ANS: (i) Those who stretch may do so because they have pre-existing injuries, or (ii) runners who stretch may be more competitive, resulting in more injuries, or some other explanation. (10 points)

5. ( 10 points) If a data distribution is symmetric, you know that the mean equals the median. Is the converse true: that is, if the mean of a data distribution equals its median, must the distribution be symmetric? Prove it's true or give an example to show it isn't.

ANS: Not true. E.g.: 6 data values, -4, -2, -1, 1, 3, 3. Median=mean=0.