A manufacturer of band saws wants to estimate the average repair cost per month

Question

A manufacturer of band saws wants to estimate the average repair cost per month for the saws he has sold to certain industries. He cannot obtain a repair cost for each saw, but he can obtain the total amount spent for saw repairs and the number of saws owned by each industry. Thus, he decides to use cluster sampling, with each industry as a cluster. The manufacturer selects a simple random sample of n = 20 from the N = 96 industries he services. The data on total cost of repairs per industry' and number of saws per industry' are as given in the accompanying table. (a) Estimate the average repair cost per saw for the past month and place a bound on the error of estimation. (b) Estimate the total amount spent by the 96 industries on band saw repairs. Place a bound on the error of estimation. (c) After checking his sales records, the manufacturer finds that he sold a total of 710 band saws to these industries. Using this additional information, estimate the total amount spent on saw repairs by these industries and place a bound on the error of estimation.

Explanation / Answer

The answer is below

n mi yi mi^2 yi^2 mi*yi (yi - mi)^2 1 3 50 9 2500 150 916.2263314 Mean repair costs (myi) 19.73076923 2 7 110 49 12100 770 12100 3 11 230 121 52900 2530 49952.25 Mean saws costs (M) 6.5 4 9 140 81 19600 1260 19600 5 2 60 4 3600 120 3600 6 12 280 144 78400 3360 78400 Using N= 96 and n=20 7 14 240 196 57600 3360 57600 EE(mCong) = [{(N – n)/(Nn*M^2 )}*{(yi - myi) 2 }/(n -1)]^1/2 (96-20)/(96*20*6.5^2)*(455693.476)/20-1^1/2 8 3 45 9 2025 135 2025 EE(mcong) = 4.7403 9 5 60 25 3600 300 3600 VRmyi= EE(mcong)/myi 4.7403/6.5 10 9 230 81 52900 2070 52900 Vrmyi= 0.7293 11 8 140 64 19600 1120 19600 L = 2VRmyi= 1.4586 12 6 130 36 16900 780 16900 myi±L = 13 3 70 9 4900 210 4900 Ls = 21.18936923 14 2 50 4 2500 100 2500 Li= 18.27216923 15 1 10 1 100 10 100 16 4 60 16 3600 240 3600 Using N =710 and n=20 17 12 280 144 78400 3360 78400 EE(mCong) = [{(N – n)/(Nn*M^2 )}*{(yi - myi) 2 }/(n -1)]^1/2 (710-20)/(710*20*6.5^2)*(455693.476)/20-1^1/2 18 6 150 36 22500 900 22500 EE(mcong) = 5.252 19 5 110 25 12100 550 12100 VRmyi= EE(mcong)/myi 5.252/6.5 20 8 120 64 14400 960 14400 Vrmyi= 0.808 = 130 2565 1118 460225 22285 455693.4763 L = 2VRmyi= 1.616 myi±L = Ls = 21.34676923 Li= 18.11476923

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