Quantitative Methods in Policy Analysis

Fall 2007

Problem Set #4

1.

The U.S. Navy and Air Force have flight testing programs to assess the reliability of their long-range ballistic missiles. A total of 222 Trident C4 missiles have been launched since 1986; of these 34 were judged failures (i.e., failure to launch or an abnormal trajectory resulting in missile destruction).

A.

What is your best estimate of the reliability of the missile? What is the standard error of this estimate? Explain, in plain English, the meaning of the standard error in this case.

Note: when I ask for "plain English," We mean an explanation free of statistical jargon that can be understood by any intelligent person.

Suppose that Russia had conducted a test program with these results. Imagine you are a U.S. intelligence officer, and that you are asked to give a "conservative" (i.e., high) estimate of the reliability of the Russian missile. What would you say?

D.

Each Trident C4 missile carried, on average, five dummy warheads. In the successful launches, the miss distance (the distance between the actual impact point and the aim point) was measured for each warhead: the mean was 107 meters, the standard deviation was 55 meters. Calculate the standard error of the mean miss distance, and explain its meaning in plain English.

Note: For missile aficionados, the median miss distance is known as the “circular error probable,” or CEP. Calculations using the CEP often assume that the distribution is symmetrical, in which case the median equals the mean.

2.

The Census Bureau believes that the response rate to its mail-in questionnaire was poor in one Anacostia neighborhood. Only 812 people are listed on forms that were returned to the Bureau, but the Bureau believes that the actual population could be much larger. Interviewers scour the neighborhood and compile a list of 1,016 people living in there; of these, 588 are listed on the mail-in forms.

A.

What is the Bureau’s best estimate of the population of the neighborhood?

B.

What assumption underlies this estimate?

C.

Extra credit: use the concept of standard error to derive a 95% confidence interval for the population .

3.

The average height of U.S. women aged 18 and older is 64.1 inches, with a standard deviation of 2.6 inches.

A.

Compute the standard error of the mean height for a random sample of 42 women.

B.

The mean height of the 42 women students in this class who gave their height is 65.0 inches. What do you conclude?

C.

The mean height of the 166 women students who have completed the class survey is 65.3 inches. What do you conclude?

4.

The UMCPsalaries.xls dataset lists the salaries for all professors at UMCP as of fall 1997.

A.

Compute the the standard error of the mean salary for a random sample of 100 professors. Interpret this in plain English.

B.

What proportion of faculty are women? Compute the standard error of the percent women for a random sample of 100 participants. Interpret in plain English.

C.

Select a simple random sample of 100 participants.

There are a number of ways to do this. The simplest is to use Tools/Data Analysis/Sampling, enter the range of the salary data, check "random" and enter 100 for "number of samples. Alternately, you can insert a new column in the data set, enter =RAND() in each cell in this column, sort the data by the random numbers, and select the first 100. (Note that new random numbers will automatically be computed after you sort. If this bothers you, turn off automatic calculation [Tools/Options/Calculation/Manual] or use copy and paste special/values to fix the random values before sorting. If you turn off automatic calculation, you must press F9 to force calculation of formulas. Finally, you can also follow the directions in Keller, section 7.3, to generate 100 random numbers uniformly distributed between 2 and 1221 and then manually select these row numbers, but this is more tedious. 

D.

Compute the mean salary and the proportion of women for the random sample. Compare the sample mean and proportion to the population mean and proportion and the relevant standard errors. Interpret in plain English.