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Statistics / Statistics for Engineers and Scientists 4 / Chapter 5 / Problem 26SE

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

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A sample of seven concrete blocks had their crushing

Chapter 5, Problem 26SE

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QUESTION:

A sample of seven concrete blocks had their crushing strength measured in MPa. The results were

1367.6 1411.5 1318.7 1193.6 1406.2

1425.7 1572.4

Ten thousand bootstrap samples were generated from these data, and the bootstrap sample means were arranged in order. Refer to the smallest mean as $Y{\ }_{2}$ the second smallest as $Y{\ }_{2}$ , and so on, with the largest being Y10,000. Assume that Y50 = 1283.4, Y51 = 1283.4, Y100 = 1291.5, Y101 = 1291.5, Y250 = 1305.5, Y251 = 1305.5, Y500 = 1318.5, Y501 = 1318.5, Y9500 = 1449.7, Y9501 = 1449.7, Y9750 = 1462.1, Y9751 = 1462.1, Y9900 = 1476.2, Y9901 = 1476.2,

Y9950 = 1483.8, and Y9951 = 1483.8.

a. Compute a $95\%$ bootstrap confidence interval for the mean compressive strength, using method 1 as described on page 390 .
b. Compute a $95\%$ bootstrap confidence interval for the mean compressive strength, using method 2 as described on page 390 .
c. Compute a $99\%$ bootstrap confidence interval for the mean compressive strength, using method 1 as described on page 390 .
d. Compute a $99\%$ bootstrap confidence interval for the mean compressive strength, using method 2 as described on page 390 .

Image Text Transcription: A sample of seven concrete blocks had their crushing strength measured in MPa. The results were 1367.6 1411.5 1318.7 1193.6 1406.2 1425.7 1572.4 Ten thousand bootstrap samples were generated from these data, and the bootstrap sample means were arranged in order. Refer to the smallest mean as Y2 the second smallest as Y2, and so on, with the largest being Y10,000. Assume that Y50 = 1283.4, Y51 = 1283.4, Y100 = 1291.5, Y101 = 1291.5, Y250 = 1305.5, Y251 = 1305.5, Y500 = 1318.5, Y501 = 1318.5, Y9500 = 1449.7, Y9501 = 1449.7, Y9750 = 1462.1, Y9751 = 1462.1, Y9900 = 1476.2, Y9901 = 1476.2, Y9950 = 1483.8, and Y9951 = 1483.8. a. Compute a 95% bootstrap confidence interval for the mean compressive strength, using method 1 as described on page 390 . b. Compute a 95%bootstrap confidence interval for the mean compressive strength, using method 2 as described on page 390 . c. Compute a 99% bootstrap confidence interval for the mean compressive strength, using method 1 as described on page 390 . d. Compute a 99% bootstrap confidence interval for the mean compressive strength, using method 2 as described on page 390 .

Questions & Answers

QUESTION:

A sample of seven concrete blocks had their crushing strength measured in MPa. The results were

1367.6 1411.5 1318.7 1193.6 1406.2

1425.7 1572.4

Ten thousand bootstrap samples were generated from these data, and the bootstrap sample means were arranged in order. Refer to the smallest mean as $Y{\ }_{2}$ the second smallest as $Y{\ }_{2}$ , and so on, with the largest being Y10,000. Assume that Y50 = 1283.4, Y51 = 1283.4, Y100 = 1291.5, Y101 = 1291.5, Y250 = 1305.5, Y251 = 1305.5, Y500 = 1318.5, Y501 = 1318.5, Y9500 = 1449.7, Y9501 = 1449.7, Y9750 = 1462.1, Y9751 = 1462.1, Y9900 = 1476.2, Y9901 = 1476.2,

Y9950 = 1483.8, and Y9951 = 1483.8.

a. Compute a $95\%$ bootstrap confidence interval for the mean compressive strength, using method 1 as described on page 390 .
b. Compute a $95\%$ bootstrap confidence interval for the mean compressive strength, using method 2 as described on page 390 .
c. Compute a $99\%$ bootstrap confidence interval for the mean compressive strength, using method 1 as described on page 390 .
d. Compute a $99\%$ bootstrap confidence interval for the mean compressive strength, using method 2 as described on page 390 .

Image Text Transcription: A sample of seven concrete blocks had their crushing strength measured in MPa. The results were 1367.6 1411.5 1318.7 1193.6 1406.2 1425.7 1572.4 Ten thousand bootstrap samples were generated from these data, and the bootstrap sample means were arranged in order. Refer to the smallest mean as Y2 the second smallest as Y2, and so on, with the largest being Y10,000. Assume that Y50 = 1283.4, Y51 = 1283.4, Y100 = 1291.5, Y101 = 1291.5, Y250 = 1305.5, Y251 = 1305.5, Y500 = 1318.5, Y501 = 1318.5, Y9500 = 1449.7, Y9501 = 1449.7, Y9750 = 1462.1, Y9751 = 1462.1, Y9900 = 1476.2, Y9901 = 1476.2, Y9950 = 1483.8, and Y9951 = 1483.8. a. Compute a 95% bootstrap confidence interval for the mean compressive strength, using method 1 as described on page 390 . b. Compute a 95%bootstrap confidence interval for the mean compressive strength, using method 2 as described on page 390 . c. Compute a 99% bootstrap confidence interval for the mean compressive strength, using method 1 as described on page 390 . d. Compute a 99% bootstrap confidence interval for the mean compressive strength, using method 2 as described on page 390 .

ANSWER:

Solution

Step 1 of 4

a) Here we have to find the 95% confidence interval for mean of bootstrap by using method-1

Let $Y{\ }_{\alpha{}/2}$ represents the 100 $\alpha{}/2$ percentile

Let $Y{\ }_{1-\alpha{}/2}$ represents the 100(1- $\alpha{}/2)$ percentile

The 95% confidence intervals are

$(Y{\ }_{\alpha{}/2}$ , $Y{\ }_{1-\alpha{}/2}$ )=( $Y{\ }_{2.5\%}$ , $Y{\ }_{97.5\%}$ )

=( $\frac{Y{\ }_{250}+Y{\ }_{251}}{2},\frac{Y{\ }_{9750}+Y{\ }_{9751}}{2}\ )$

= $(\frac{1305.5+1305.5}{2},\ \frac{1462.1+1462.1}{2})$

= (1305.5, 1462.1)

The 95% confidence interval using method 1 is (1305.5, 1462.1)

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