STAT-110

52. About 20% of the engineering students at a large university are women. The school plans to poll a sample of 200 engineering students about the quality of student life.
    1. If an SRS of size 200 is selected, about how many women do you expect to find in the sample?

       I'd expect to find about 20% of the sample (40 women).

    2. If the poll wants to be able to report separately the opinions of male and female students, what type of sampling design would you suggest? Why?

       I'd choose a stratified sample, so that the number of men and women in the sample can be specified and adequate precision obtained for both groups. If we knew how many women and men were engineering students, we could use a probability sample.

58. A city contains 33 supermarkets. A health inspector wants to check compliance with a new city ordinance on meat storage. Because of the time required, he can inspect only 10 markets. He decides to choose a stratified random sample and stratifies the markets by sales volume. Stratum A consists of 3 large chain stores; the inspector decides to inspect all 3. Stratum B consists of 10 smaller chain stores; 4 out of the 10 will be inspected. Stratum C consists of 20 locally owned small stores; 3 of these 20 will be inspected. Let "Yes" mean that the store is in compliance and "No" mean that it is not. The population is as follows (unknown to the inspector, of course):

    Stratum A			Stratum B			Stratum C
    1	Yes		1	No		01	Yes		11	Yes
    2	Yes		2	Yes		02	Yes		12	Yes
    3	No		3	No		03	No		13	No
    			4	No		04	Yes		14	Yes
    			5	Yes		05	No		15	Yes
    			6	No		06	No		16	No
    			7	Yes		07	No		17	No
    			8	No		08	Yes		18	No
    			9	No		09	No		19	Yes
    			0	Yes		10	No		20	Yes

    1. Use Table A to choose a stratified random sample of size 10 allotted among the strata as described above.

       We'll choose stores 1, 2, 3 from Stratum A.

       Using the numbering above for Stratum B, we'll use Line 110 to select our sample.
       Our selections are: 3|8|4|4|8 | 4|8|7|8|9
       Therefore, we'll choose stores 3, 4, 7, 8 from Stratum B.

       Using the numbering above for Stratum C, we'll use Line 131 to select our sample.
       Our selections are: 05|00|7 1|66|32 | 81|19|4 1|48|37 | 04|19|7 8|55|76
       Therefore, we'll choose stores 04, 05, 19 from Stratum C.

    2. Use your sample results to estimate the proportion of the entire population of stores in compliance with the ordinance.

       Stratum A: Yes, Yes, No (²/₃ of stores comply)
       Stratum B: No, No, Yes, No (¹/₄ of stores comply)
       Stratum C: Yes, No, Yes (²/₃ of stores comply)

       So, the estimated number in compliance (using the properties of a probability sample) is:

       ²/₃(3) + ¹/₄(10) + ²/₃(20) = 2 + 2.5 + 13.3333 = 17.8333 or about 18 stores.

       Therefore, the estimated proportion in compliance (a statistic) is:

       ¹⁸/₃₃ = ⁶/₁₁ or about 0.54545 = 54.545%

    3. Use the description of the population given above to find the true proportion of stores in compliance. How accurate is the estimate from part (b)?

       In the population, 16 of the 33 stores are in compliance, so the true proportion of stores in compliance (a parameter) is:

       ¹⁶/₃₃ which about 0.4848 = 48.48%

       This accuracy is not very high, and it will vary from sample to sample (depending on how you numbered the stores and which lines of Table A you used). The estimation procedure is unbiased, but has low precision due to the small sample size.

Please direct all questions regarding STAT-110 to your instructor or to the director of STAT-110, Dr. Todd Ogden at ogden@stat.sc.edu.

Mail comments regarding this presentation to W. Scott Street, IV at street@stat.sc.edu.

© 1996 by W. Scott Street, IV