36-303: Sampling, Surveys & Society HW05: Hand in on paper Thu Apr 07, 2011

\underline{Reminders:}

• Please see Lecture 18, Week 09, for an outline of events for the rest of the semester.

• Reading:

– Hill Handout on Oral Presenations

– Bem Handout on Writing Papers

Examples of past 303 papers, are on the course website, under the link Handouts on Writing & Speaking.

• Exercises #2 and #3 depend on material I will present in lecture Mar 31 and Apr 05.

Exercises to Turn In (there are 3 exercises):

1. Cluster Sampling (Lohr, 1999, p. 170). An accounting firm is interested in estimating the error rate in a compliance audit it is conducting. The population consists of 828 claims and the firm audits an SRS of 85 of those claims. In each of the 85 sampled claims, 215 fields are checked for errors. We can think of the claims as psu’s (clusters) and the fields as ssu’s (units), in a single-stage cluster sample.

Several clusters had the same number of errors, so the results of this single-stage cluster sample can be summarized as follows:

Number of claims Number of errors  \overline{y}_{i}
(clusters)  out of 215 fields = (#errors)/(#fields)
1  4  0.01860
1  3  0.01395
4  2  0.00930
22  1  0.00465
57  0  0.00000

(data courtesy of Fritz Scheuren).

(a) Use the ideas about cluster sampling discussed in class and in the handout on cluster sampling, to estimate the proportion of all 828×215=178,020 fields with errors in them. Give a standard error for your estimate.

(b) Use your answer in part (a) to estimate, and give a standard error for, the total number of errors in all 178,020 fields.

(c) Suppose that, instead of taking a cluster sample, the firm takes an SRS of (85) · (215) = 18, 275 fields from the 178, 020 fields in the population. If the estimated error rate is the same as in part (a), what would the estimated variance be? How does this compare to the estimated variance in part (a)?

2. Taylor Series Variance Estimates. At a certain university there are 1998 undergraduates. An SRS without replacement, of size n = 20 is taken, to find out what is the average number of hours per week the students study. Some demographic data was collected also. Here is the data from 20 respondents in the survey:

Sex College Hrs/Wk
M  Eng  28
M  Eng  29
M  Eng  23
M  Eng  35
M  Eng  29
M  Eng  30
M  Eng  34
M  Eng  31
F  Eng  30
F  Eng  31

Sex College Hrs/Wk
F  Eng  36
F  Eng  33
M  Lib  27
M  Lib  28
F  Lib  29
F  Lib  30
F  Lib  28
F  Lib  28
F  Lib  32
F  Lib  30

(a) Estimate the mean number of hours/wk students study at this university, and give the standard error.

(b) Administrative records show that the total undergrad enrollment at the university is

Sex Eng Lib
M 617 380
F 450 551

Make four post-strata by combining sex and college of the students, compute post-stratum weights to make the sample more representative of the population, and compute the weighted average number of hours/wk students study.

(c) Use the Taylor Series method discussed in class to compute a standard error for your weighted estimate in part (b). Compare to the standard error you computed in part (a).

3. Jackknife Variance Estimates. Use the jackknife method discussed in class to compute a standard error for the weighted estimate in problem 2(b). Compare to the standard errors you calculated in problems 2(a) and 2(c).

\underline{Notes on #2 and #3} : The sample size n = 20 is small enough that you can do all the computations “by hand” using Minitab, R, Excel, etc. If you know R and you want to compare your answers with the results you get from the R functions I introduced in class, please feel free to do so.