Mont4e_sm_ch08_sec06 - Section 8-7 8-55 95 prediction interval on the life of the next tire given x = 60139.7 s = 3645.94 n = 16 for =0.05 t/2,n-1

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Unformatted text preview: Section 8-7 8-55 95% prediction interval on the life of the next tire given x = 60139.7 s = 3645.94 n = 16 for =0.05 t/2,n-1 = t0.025,15 = 2.131 x - t 0.025,15 s 1 + 60139.7 - 2.131(3645.94) 1 + 1 1 x n +1 x + t 0.025,15 s 1 + n n 1 1 x n +1 60139.7 + 2.131(3645.94) 1 + 16 16 52131.1 x n +1 68148.3 The prediction interval is considerably wider than the 95% confidence interval (58,197.3 62,082.07). This is expected because the prediction interval needs to include the variability in the parameter estimates as well as the variability in a future observation. 8-56 99% prediction interval on the Izod impact data 8-20 n = 20 x = 1.25 s = 0.25 t 0.005,19 = 2.861 x - t 0.005,19 s 1 + 1.25 - 2.861(0.25) 1 + 1 1 x n +1 x + t 0.005,19 s 1 + n n 1 1 x n +1 1.25 + 2.861(0.25) 1 + 20 20 0.517 x n +1 1.983 The lower bound of the 99% prediction interval is considerably lower than the 99% confidence interval (1.108 ). This is expected because the prediction interval needs to include the variability in the parameter estimates as well as the variability in a future observation. 8-57 95% Prediction Interval on the volume of syrup of the next beverage dispensed x = 1.10 s = 0.015 n = 25 t/2,n-1 = t0.025,24 = 2.064 x - t 0.025, 24 s 1 + 1.10 - 2.064(0.015) 1 + 1 1 x n +1 x + t 0.025, 24 s 1 + n n 1 1 x n +1 1.10 - 2.064(0.015) 1 + 25 25 1.068 x n +1 1.13 The prediction interval is wider than the confidence interval:1.094 8-58 1.106 90% prediction interval the value of the natural frequency of the next beam of this type that will be tested. given x = 231.67, s =1.53 For = 0.10 and n = 5, t/2,n-1 = t0.05,4 = 2.132 x - t 0.05, 4 s 1 + 231.67 - 2.132(1.53) 1 + 1 1 x n +1 x + t 0.05, 4 s 1 + n n 1 1 x n +1 231.67 - 2.132(1.53) 1 + 5 5 228.1 x n +1 235.2 The 90% prediction in interval is greater than the 90% CI. 8-59 95% Prediction Interval on the volume of syrup of the next beverage dispensed n = 20 x = 485.8 s = 90.34 t/2,n-1 = t0.025,19 = 2.093 x - t 0.025,19 s 1 + 485.8 - 2.093(90.34) 1 + 1 1 x n +1 x + t 0.025,19 s 1 + n n 1 1 x n +1 485.8 - 2.093(90.34) 1 + 20 20 292.049 x n +1 679.551 8-60 The 95% prediction interval is wider than the 95% confidence interval. 99% prediction interval on the polyunsaturated fat 8-21 n = 6 x = 16.98 s = 0.319 t 0.005,5 = 4.032 x - t 0.005,5 s 1 + 16.98 - 4.032(0.319) 1 + 1 1 x n +1 x + t 0.005,5 s 1 + n n 1 1 x n +1 16.98 + 4.032(0.319) 1 + 6 6 15.59 x n +1 18.37 The length of the prediction interval is much longer than the width of the confidence interval 16.455 17.505 . 8-61 Given x = 317.2 s = 15.7 n = 10 for =0.05 t/2,n-1 = t0.005,9 = 3.250 x - t 0.005,9 s 1 + 317.2 - 3.250(15.7) 1 + 1 1 x n +1 x + t 0.005,9 s 1 + n n 1 1 x n +1 317.2 - 3.250(15.7) 1 + 10 10 263.7 x n +1 370.7 The length of the prediction interval is longer. 8-62 95% prediction interval on the next rod diameter tested n = 15 x = 8.23 s = 0.025 t 0.025,14 = 2.145 x - t 0.025,14 s 1 + 1 1 x n +1 x + t 0.025,14 s 1 + n n 8.23 - 2.145(0.025) 1 + 1 1 x n +1 8.23 - 2.145(0.025) 1 + 15 15 8.17 x n +1 8.29 95% two-sided confidence interval on mean rod diameter is 8.216 8.244 8-63 90% prediction interval on the next specimen of concrete tested given x = 2260 s = 35.57 n = 12 for = 0.05 and n = 12, t/2,n-1 = t0.05,11 = 1.796 x - t 0.05,11 s 1 + 2260 - 1.796(35.57) 1 + 1 1 x n +1 x + t 0.05,11 s 1 + n n 1 1 x n +1 2260 + 1.796(35.57) 1 + 12 12 2193.5 x n +1 2326.5 8-64 90% prediction interval on wall thickness on the next bottle tested. 8-22 given x = 4.05 s = 0.08 n = 25 for t/2,n-1 = t0.05,24 = 1.711 x - t 0.05, 24 s 1 + 4.05 - 1.711(0.08) 1 + 1 1 x n +1 x + t 0.05, 24 s 1 + n n 1 1 x n +1 4.05 - 1.711(0.08) 1 + 25 25 3.91 x n +1 4.19 8-65 90% prediction interval for enrichment data given x = 2.9 s = 0.099 n = 12 for = 0.10 and n = 12, t/2,n-1 = t0.05,11 = 1.796 x - t 0.05,12 s 1 + 2.9 - 1.796(0.099) 1 + 1 1 x n +1 x + t 0.05,12 s 1 + n n 1 1 x n +1 2.9 + 1.796(0.099) 1 + 12 12 2.71 x n +1 3.09 The 90% confidence interval is x - t 0.05,12 s 2.9 - 1.796(0.099) 1 1 x + t 0.05,12 s n n 1 1 2.9 - 1.796(0.099) 12 12 2.85 2.95 The prediction interval is wider than the CI on the population mean with the same confidence. The 99% confidence interval is x - t 0.005,12 s 2.9 - 3.106(0.099) 1 1 x + t 0.005,12 s n n 1 1 2.9 + 3.106(0.099) 12 12 2.81 2.99 The prediction interval is even wider than the CI on the population mean with greater confidence. 8-66 To obtain a one sided prediction interval, use t,n-1 instead of t/2,n-1 Since we want a 95% one sided prediction interval, t/2,n-1 = t0.05,24 = 1.711 and x = 4.05 s = 0.08 n = 25 x - t 0.05, 24 s 1 + 4.05 - 1.711(0.08) 1 + 1 x n +1 n 1 x n +1 25 3.91 x n +1 The prediction interval bound is much lower than the confidence interval bound of 8-23 4.023 mm ...
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This note was uploaded on 10/12/2009 for the course IND E 315 taught by Professor Kailashkapur during the Spring '09 term at University of Washington.

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