Quiz #2

1. (4.51) Enzyme-linked immunosorbent assay (ELISA) is the most common type of screening test for detecting the HIV virus. A positive result from and ELISA has a high degree of sensitivity (to detect infection) and specificity (to detect non-infection). Suppose the probability that a person is infected with the HIV virus for a certain population is 0.015. If the HIV virus is actually present, the probability that the ELISA test will give a positive result is 0.995. If the HIV virus is not actually present the probability of a positive result from and ELISA is 0.01. If the ELISA has given a positive result, use Bayes’ theorem to find the probability that the HIV virus is actually present.

2. (5.3) Recently, a regional automobile dealership sent out fliers to perspective customers, indicating that they had already won one of three different prizes: a 2008 Kia Optima valued at $15,000, a $500 gas card, or a $5 Wal-Mart shopping card. To claim his or her prize, a prospective customer needed to present the flier at the dealership’s showroom. The fine print on the back of the flier listed the probabilities of winning. The chance of winning the car was 1 out of 31,478, the chance of winning the gas card was 1 out of 31,478 and the chance of winning the shopping card was 31,476 out of 31,478.

a. How many fliers do you think the automobile dealership sent out?

b. Using your answer to (a) and the probabilities listed on the flier, what is the expected value of the prize won by a prospective customer receiving a flier?

c. Using your answer to (a) and the probabilities listed on the flier, what is the standard deviation of the value of the prize won by a prospective customer receiving a flier?

3. (6.3) Given a standardized normal distribution (as in table E.2) determine the following probabilities:

a. Z is less than 1.08?

b. Z is greater than -0.21?

c. Z is less than -0.21 or greater than the mean?

d. Z is less than -0.21 or greater than 1.08?

4. (6.13) Many manufacturing problems involve the matching of machine parts, such as shafts that fit into a valve hole. A particular design requires a shaft with a diameter of 22,000 mm, but shafts with diameters between 21.990 mm and 22.010 mm are acceptable. Suppose that the manufacturing process yields shafts with diameters normally distributed, with a mean of 22.002 mm and a standard deviation of 0.005 mm. For this process, what is

a. the proportion of shafts with a diameter between 21.99 mm and 22.00 mm?

b. the probability that a shaft is acceptable?

c. the diameter that will be exceeded by only 2% of the shafts?

5. (6.21) The file SavingsRate contains the yields for a money market account, a one-year certificate of deposit (CD), and a five-year CD for 23 Banks in the metropolitan New York area, as of May 28, 2009. For each of the three types of investments decide whether the data appear to be approximately normally distributed by

a. comparing data characteristics to theoretical properties

b. constructing a normal probability plot.