Elements of Statistics (Math 106) – Exam 2
Fall 2005 – Brad Hartlaub
Name:________________________
Directions: Please answer all six questions and show your work. The point values for each
problem are indicated in parentheses. You may use one sheet of formulas and any software that
is available on the Kenyon network during the exam. Good luck and have a nice break!
1. For the population of people who suffer occasionally from migraine headaches, suppose p =
0.6 is the proportion who get some relief from taking ibuprofen. A random sample of 24 people
who suffer from migraines is selected. Let X denote the number of people in the sample who
experienced relief after taking ibuprofen.
a.) Identify the probability distribution of X. (5)
Find the following probabilities: (15, 3 each part)
b.) P  X  12 
c.) P  X  12
d.) P  X  12
e.) P  X  16
f.) P  X  16
2. The Eurobarometer survey has tracked opinions of Europeans about the common currency
(the euro) that is now used in many European countries. When it was introduced in January,
2002, 67% of adult residents of the affected countries indicated that they were happy that the
euro had arrived. Suppose a poll of size 1000 is planned next year to estimate the percentage of
people who now approve of the common currency. Suppose the population proportion still
equals 0.67.
a.) Identify the mean and standard deviation of the statistic of interest to the researchers.
(5)
b.) Suppose that 800 people in the poll indicate approval. Does this give strong evidence
that the percentage approving of the euro has gone up? Why or why not? Justify
your response with an appropriate probability calculation. (10)
3. Vincenzo De Cerce was diagnosed with high blood pressure. He was able to keep his blood
pressure under control for several months by taking blood pressure medicine (amlodipine
desylate). Mr. De Cerce’s blood pressure is monitored by taking 3 readings a day, in the early
morning, at mid-day, and in the evening.
a.) During this period, suppose that the probability distribution of his systolic blood
pressure reading had a mean of 130 and a standard deviation of 6. If the successive
observations behave like a random sample from such a distribution, find the mean and
standard deviation of the sampling distribution of the sample mean for each day. (5)
b.) Make the additional assumption that the probability distribution of his systolic blood
pressure reading is normal. Find the probability that the sample mean exceeds 140,
which is considered excessively high. (10)
4. The state of Ohio has several statewide lottery options. One of the simpler ones is the “Pick
3” game in which you pick one of the 1000 3-digit numbers between 000 and 999. The lottery
selects a 3-digit number at random. For a bet of $1, you win $500 if your number is selected and
nothing ($0) otherwise.
a.) With a single $1 bet, what is the probability that you win $500? (5)
b.) Let X denote your winnings on a $1 bet. Construct the probability distribution for X.
(5)
c.) The mean of X, X = 0.50. Provide an interpretation of this value. (5)
d.) If you play in two different games, find the probability that you win both times. (5)
e.) The profit Y from buying a $1 ticket equals the winnings X minus the dollar paid for
the ticket. Would you expect the standard deviation of the distribution of Y to be
equal to, larger than, or smaller than the standard deviation of the distribution of X?
Explain. (10)
f.) If you play this game five different times, what are the mean and standard deviation of
your total winnings? (Hint: X = 0.50, and X = 15.8). (10)
5. The Major Histocompatability Complex (MHC) is an important set of genes that influences
both immune function and body odor in mammals, including humans. Evidence suggests that
mice and other mammals prefer mates that are genetically dissimilar in their MHC, because
“mixed” offspring have stronger immune systems. In 1995, evolutionary biologists conducted an
experiment to see if MHC could influence human mate choice. They had women smell t-shirts
that had been slept in for two nights by 38 different men. Each man’s shirt was sniffed by two
groups of women: MHC1 and MHC2, who rated the “pleasantness” of the t-shirt odor on a scale
from 0 – 10, with 5 being neutral. The MHC2 group was more similar to the men in terms of
genotype, so the researchers hypothesized that MHC1 women should find the odors more
pleasant than the MHC2 women.
When smelled by the MHC1 women, the 95% C.I. for the mean pleasantness rating was
5.941.71. When smelled by MHC2 women, the 95% C.I. was 4.631.26.
a.) Does the 95% C.I. for the MHC2 women contain the sample mean rating of the MHC1
women? Justify your response. (5)
b.) Based on two-tailed hypothesis tests at the  = 0.05 significance level, can either group’s
rating be distinguished from 5 (or neutral)? Explain how you know. (10)
c.) Based on the hypothesis that MHC1 women should find a man’s scent more pleasant than
MHC2 women, select the appropriate hypothesis test from the choices below. (5)
a.
b.
c.
d.
e.
One-tailed test, HA: MHC1 = MHC2
One-tailed test, HA: MHC1 < MHC2
One-tailed test, HA: MHC1 > MHC2
Two-tailed test, HA: MHC1 > MHC2
Two-tailed test, HA: MHC1 ≠ MHC2
6. Polychlorinated biphenyls (PCBs) are man-made pollutants that can cause harmful health
effects when consumed by humans and other animals. Because PCBs can accumulate in fish, the
US Environmental Protection Agency has conducted comprehensive surveys of fish tissue PCB
concentrations in lakes around the country. Data detailing the concentrations of two of the more
common PCBs (138 and 153) as well as total PCB concentration (in parts per billion, ppb) in fish
from 68 lakes are available in P:/Data/MATH/STATS/pcb2.mtw. Assume that this is a SRS.
a.) Find and interpret a 95% confidence interval for the mean total PCB concentration. (10)
b.) Suppose that fish with total PCB concentrations higher than 55 ppb are considered unsafe
for human consumption. Is the mean total fish PCB concentration in US lakes higher than
55? State the null and alternative hypotheses, perform the appropriate hypothesis test at
the  = 0.01 significance level, and interpret your results. (15)
c.) Do the mean concentrations of PCBs 138 and 153 differ? State the appropriate null and
alternative hypotheses, conduct the appropriate hypothesis test at the  = 0.05
significance level, and interpret your results. (15)