Quantitative Methods in Policy Analysis

Fall 2007

Problem Set #3

Hurricane.xls contains data on the number of hurricanes that made landfall in the United States in each year, from 1887 to 2005. 

Using the Poisson distribution, calculate the probability of 0, 1... 7 hurricanes making landfall in the United States in a given year, assuming that hurricanes are independent random events with the mean frequency of 1.7 hurricanes per year. Compare the expected probabilities and frequencies with the observed probabilities and frequencies in part A. Is the distribution of hurricanes making landfall each year consistent with the assumption of independent random events?

2.

Nationwide scores on the verbal part of the GRE are normally distributed with a mean of 480 and a standard deviation of 95.

A.

A student receives a score of 580. What is the corresponding percentile rank?

B.

A student tells you they scored in the 23rd percentile. What score does this correspond to?

3.

Lead toxicity affects an estimated 5 percent of preschool children. A new, less painful and less expensive screening test is proposed to determine whether children have lead toxicity. A sample of 100 children known not to have lead toxicity and 100 children known to have lead toxicity were recruited for a preliminary study of the effectiveness of the test. For the children without lead toxicity, the mean test value was 39 and the standard deviation was 13; for the children with lead toxicity, the mean was 95 and the standard deviation was 27. The test values for each group were approximately normally distributed.  (Although it isn't important, you can think of these test values as the measured concentration (e.g., in micrograms per deciliter) of a substance in the blood.)

A.

What percentage of children with lead toxicity had test values above 95? above 60? below 60?

B.

For children with lead toxicity, what test value corresponds to the 5^th percentile (i.e., 5 percent of children have lower values, 95% have higher values)?

C.

The answer to part B can be considered a threshold value for lead toxicity; in other words, children with values higher than this threshold could be considered "positive" for lead toxicity. What percentage of children without lead toxicity had test values above this threshold?

D.

Based on your answers to B and C, what are the false positive and false negative rates for the test?

E.

If a child has a positive test, what is the probability that the child has lead toxicity?

F.

Comment on the usefulness of this test as a screening tool for the general population. In what situations would this test be more useful?