An economic survey is designed to estimate the average amount spent on utilities

Question

An economic survey is designed to estimate the average amount spent on utilities for households in a city. Because no list of households is available, cluster sampling used, with divisions (wards) forming the clusters. A simple random sample of 20 wards is selected from 60 wards of city. Interviewers then obtain the cost of utilities from each household within the sampled wards; the total costs are shown in accompanying numbers:

Sampled Ward: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20

Number of Households: 55,60,63,58,71,78,69,58,52,71,73,64,69,58,63,75,78,51,67,70  

Total Amount Spent on Utilities ($): 2210,2390,2430,2380,2760,3110,2780,2370,1990,2810,2930,2470,2830,2370,2390,2870,3210,2430,2730,2880

a. Estimate the average amount a household in the city spends on utilities, and place a bound on the error of estimation.

b. In survey, the number of households in the city is not known. Estimate the total amount spent on utilities for all households in the city and place a bound on the error of estimation.

c. The economic survey is to be performed in a neighboring city of similar structure. The objective is to estimate the total amount spent on utilities by households in the city, with a bound of $5000 on the error of estimation. Use the data above to find the appropriate number of clusters needed to achieve this bound.  

Explanation / Answer

a.

Average amount a household spends in the city on utilities = 40.15

Error of estimation = 0.459

b.

Amount spent on utilities for 20 wards = 52340

Average amount spent on utilities per ward = 52340/20 = 2617

Total amount spent on utilities for all households in the city = 2617*60 = 157020

Error of estimation = 70.70732859*60 = 4242.44

c.

Error of estimation = St. deviation/square root of n

=> 5000 = 60*316.212/sqrt(n)

=> n = 15

15 Clusters required

Ward No. No. of Households Total amount spent Average amount spent per household 1 55 2210 40.18181818 2 60 2390 39.83333333 3 63 2430 38.57142857 4 58 2380 41.03448276 5 71 2760 38.87323944 6 78 3110 39.87179487 7 69 2780 40.28985507 8 58 2370 40.86206897 9 52 1990 38.26923077 10 71 2810 39.57746479 11 73 2930 40.1369863 12 64 2470 38.59375 13 69 2830 41.01449275 14 58 2370 40.86206897 15 63 2390 37.93650794 16 75 2870 38.26666667 17 78 3210 41.15384615 18 51 2430 47.64705882 19 67 2730 40.74626866 20 70 2880 41.14285714 Aggrerate 1303 52340 40.16884114 Standard Deviation 316.2127865 2.051682323 Margin of error 70.70732859 0.458770114

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