1. (44 pts) Scientists are investigating to see whether or not there is a
systematic difference in the weights obtained with two scales.

      WEIGHT IN HUNDREDTHS OF GRAMS

Rock specimen     Scale I     Scale II    Difference

  1           1213            1217         -4
  2           1756            1761         -5
  3            933              935         -2
  4           1140            1142         -2
  5           2862            2861          1
  6           1025            1027         -2
  7           2337            2342         -5
  8           1627            1626          1
  9               1240           1245             -5
  10              2478        2475          3
 
MEAN              1661.10     1663.10             -2
STD DEV            679.94     678.73              2.87

a. (03 pts) Compute the STANDARD ERROR OF THE DIFFERENCE between the Scale
I and the Scale II means.

b. (08 pts) Compute a 90% confidence interval for the difference between
the Scale I and the Scale II means. Show all steps, including writing down
the tabled value that you need to obtain from Zar in order to do the
confidence interval. Your "final answer" should be in the form: [lower
number, upper number].

c. (04 pts) Use your result from part (b.) to answer the following
question: "On average, do the two scales yield the same result?", and give
the reason why.  In answering this question using the confidence interval
from (b.), what is the probability of a Type I Error here?

d. (08 pts) Compute the statistical power of being able to detect a
difference of 1.0, on average, between the two scales, at a .10 level of
significance, using 10 rock specimens. Use ONLY the tables in Zar to
obtain your final power statement. (The purpose of this is to ensure that
you know how to read tables in Zar.) Your "final answer should be in the
form: [lower number for power, upper number for power].

e. (21 pts) If we didn't want to assume normality, do the nonparametric
test at the .10 level to see if there is a difference between the two
scales regarding weights. Write down your null and alternative hypotheses,
show all steps in computing the test statistic, decision rule, and be sure
to include the P-value associated with the observed test statistics. What
is your conclusion?

2. (31 pts) Scientists have obtained the grain size (diameters in
hundredths of millimeters) of grains of sand taken from two different
locations on the moon.

Location 1: 40   73   73   27   45   23   62

Location 2: 86   47   80   44   84   27   >100*

*NOTE: one of the grains exceeded the capacity of the measuring
instrument.

At the .05 level of significance, use the best available test to test
whether the two populations are the same, or whether one is shifted from
the other. Include the following in your write-up: null and alternative
hypotheses, computed test statistic showing all steps, tabled critical
value from Zar, decision rule, P-value associated with your computed test
statistic (using ONLY Zar table to obtain this), and your final
conclusion.

3. (54 pts). Six guinea pigs (the "A" Group) were injected with 0.5 mg of
a medication.  Five other guinea pigs (the "B" group) were injected with
1.5 mg of the same medication. For each guinea pig, the "time to fall
asleep" (determined by noting certain drooping of the eyelids) was
recorded. The summary data are as follows:

"A" mean = 15.4 seconds, std. deviation = 2.2 seconds, n=6
"B" mean = 10.6 seconds, std. deviation = 2.6 seconds, n=5

a. (14 pts) Using the .10 level of significance, test to see whether the
population variances of the "A" group and the "B" group may be considered
the same. Include the following in your write-up: null and alternative
hypotheses, computed test statistic showing all steps, tabled critical
value from Zar, decision rule, P-value associated with computed your test
statistic (using ONLY Zar table to obtain P-value), and your final
conclusion.

b. (10 pts) Using your result from part (a.), compute the STANDARD ERROR
OF THE DIFFERENCE between the "A"  observed mean and the "B" observed
mean.

c. (15 pts) Do a parametric test, at the .05 level of significance, to
test whether or not the increase in dosage will REDUCE the average time it
takes a guinear pig to fall asleep. Include the following in your
write-up: null and alternative hypotheses, computed test statistic showing
all steps, tabled critical value from Zar, decision rule, P-value
associated with your computed test statistic (using ONLY Zar table to
obtain this), and your final conclusion.

d. (15 pts) In a future experiment, how large a total sample size,
assuming equal sample sizes for the two groups, would be needed to detect
a mean reduction in time-to-fall-asleep of 2.0 seconds? Use .05 level of
significance, 90% power, and best available variance estimate from part
(b.) or (c.).