Exam #2 – Math 106



Elements of Statistics (Math 106) – Exam 2 Name:________________________

Fall 2005 – Brad Hartlaub

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)

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c.) [pic]

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e.) [pic]

f.) [pic]

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. One-tailed test, HA: (MHC1 = (MHC2

b. One-tailed test, HA: (MHC1 < (MHC2

c. One-tailed test, HA: (MHC1 > (MHC2

d. Two-tailed test, HA: (MHC1 > (MHC2

e. 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)

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