Deck 30: The Bootstrap

Question 1

Consider the following sample of three measurements {4,8,13}.How many distinct resamples are possible?

A)27
B)24
C)3
D)6
E)10

Answer 1

10

Question 2

Consider the following sample of three measurements {4,8,13}.What is the probability of observing the distinct resample {4,8,8}?

A)3/8
B)1/8
C)1/4
D)1/27
E)1/9

Answer 2

1/9

Question 3

A random sample of 50 adult males in Toronto,Ontario were randomly selected and their weight,in kilograms,was recorded to the nearest kilogram.In order to compute a bootstrap confidence interval for the median,400 subsequent bootstrap samples were taken and the resulting sample medians were ordered from lowest to highest.The first 10 and last 10 ordered medians are given below: First 10: 68.0,68.5,68.5,69.0,69.5,70.0,70.0,70.0,70.5,70.5
Last 10: 93.5,94.0,94.0,94.5,94.5,94.5,95.0,95.5,95.5,96.0
Use the percentile method to compute a 98% bootstrap confidence interval for the true median weight of adult males in Toronto,Ontario.

A)(68.5,95.0)
B)(70.5,93.5)
C)(68.0,96.0)
D)(69.0,95.0)
E)(68.0,95.0)

Answer 3

(69.0,95.0)

Question 4

Consider the following sample of four measurements {1,2,3,4}.Which of the following distinct resamples is least likely to occur?

A){1,1,1,1}
B){1,2,4,4}
C){1,1,2,2}
D){1,1,1,4}
E){1,2,3,4}

Question 5

A random sample of 50 adult men in Toronto were randomly selected in order to investigate the correlation between their heights and weights.In order to compute a bootstrap confidence interval for the difference in population means,280 subsequent bootstrap samples were taken and the resulting values of the sample correlation were ordered from lowest to highest.The first 10 and last 10 ordered values of sample correlation are given below: First 10: 0.23,0.26,0.31,0.33,0.37,0.38,0.41,0.42,0.43,0.44
Last 10: 0.66,0.67,0.70,0.73,0.75,0.76,0.79,0.81,0.85,0.88
Use the percentile method to compute a 95% bootstrap confidence interval for the true correlation between the heights and weights of adult men in Toronto.

A)(0.23,0.88)
B)(0.37,0.76)
C)(0.42,0.70)
D)(0.41,0.73)
E)(0.26,0.85)

Question 6

Consider the following sample of three measurements {4,8,13}.Five bootstrap samples of this sample produced the following resamples: {4,8,8},{4,4,13},{4,8,13},{8,8,8},{8,13,13}.Based on these five resamples,find the bootstrap of the standard error of the sample mean.

A)3.4111
B)1.8469
C)2.3157
D)0.8260
E)8.2667

Question 7

Consider the following sample of three measurements {4,8,13}.Which of the following distinct resamples is least likely to occur?

A){4,8,13}
B){8,8,8}
C){4,4,13}
D){4,8,8}
E)All distinct resamples are equally likely.

Question 8

Consider the following sample of three measurements {4,8,8}.How many distinct resamples are possible?

A)24
B)27
C)3
D)10
E)4

Question 9

Consider the following sample of four measurements {1,2,3,4}.Which of the following distinct resamples is most likely to occur?

A){1,1,1,4}
B){1,2,3,4}
C){1,2,4,4}
D){1,1,1,1}
E){1,1,2,2}

Question 10

A random sample of 100 students were randomly selected from your school and their average grade was recorded,as a percentage,for the 2010/2011 school year.In order to compute a bootstrap confidence interval for the mean,200 subsequent bootstrap samples were taken and the 200 resulting sample means were ordered from lowest to highest.The first 10 and last 10 ordered means are given below: First 10: 59.41,59.56,60.13,60.44,61.22,61.67,62.45,62.48,62.56,63.01
Last 10: 78.13,78.25,78.67,79.12,79.33,80.57,80.98,81.88,81.99,82.71
Use the percentile method to compute a 95% bootstrap confidence interval for the true average grade of all students at your school.

A)(60.44,80.98)
B)(61.67,80.57)
C)(61.22,79.33)
D)(61.22,80.57)
E)(63.01,78.13)

Question 11

Two brands of flares - Brand 1 and Brand 2 - are tested for their burning times (in minutes).A random sample of 40 of each brand of flare are burned and timed.In order to compute a bootstrap confidence interval for the difference in population means,400 subsequent bootstrap samples were taken and the resulting values of $\bar { x } _ { 1 }$ *- $\bar { x } _ { 2 }$ * were ordered from lowest to highest.The first 10 and last 10 ordered values of $\bar { x } _ { 1 }$ *- $\bar { x } _ { 2 }$ * are given below: First 10: -0.8,-0.1,0.4,0.9,1.2,1.3,1.5,1.6,1.7,1.8
Last 10: 5.1,5.5,5.9,6.1,6.3,6.8,7.0,7.5,8.1,8.8
Use the percentile method to compute a 95% bootstrap confidence interval for the true difference in the average burning time (Brand 1 - Brand 2).

A)(0.4,7.5)
B)(1.3,6.3)
C)(1.2,6.8)
D)(1.8,5.1)
E)(0.4,8.1)

Question 12

Consider the following sample of four measurements {2,5,11,14}.Five bootstrap samples of this sample produced the following resamples: {2,5,5,14},{5,11,11,11},{2,11,14,14},{2,2,14,14}, .Based on these five resamples,find the bootstrap of the standard error of the sample mean.

A)0.6452
B)4.5667
C)8.6
D)1.4427
E)2.0813

Question 13

Consider the following sample of three measurements {4,8,13}.Five bootstrap samples of this sample produced the following resamples: {4,8,8},{4,4,13},{4,8,13},{8,8,8},{8,13,13}.Based on these five resamples,find the bootstrap of the standard error of the sample median.

A)3.1937
B)1.4283
C)8.2
D)5.2222
E)10.2

Question 14

A random sample of 100 students were randomly selected from your school and their average grade was recorded,as a percentage,for the 2010/2011 school year.In order to compute a bootstrap confidence interval for the mean,200 subsequent bootstrap samples were taken and the 200 resulting sample means were ordered from lowest to highest.The first 10 and last 10 ordered means are given below: First 10: 59.41,59.56,60.13,60.44,61.22,61.67,62.45,62.48,62.56,63.01
Last 10: 78.13,78.25,78.67,79.12,79.33,80.57,80.98,81.88,81.99,82.71
Use the percentile method to compute a 90% bootstrap confidence interval for the true average grade of all students at your school.

A)(60.44,80.98)
B)(61.22,80.57)
C)(63.01,78.13)
D)(61.22,79.33)
E)(61.67,80.57)

Question 15

Consider the following sample of three measurements {4,8,13}.What is the probability of observing the distinct resample {8,8,8}?

A)1/3
B)0
C)1/27
D)1
E)1/8

Question 16

Consider the following sample of three measurements {4,8,13}.Which of the following distinct resamples is most likely to occur?

A){4,4,13}
B){8,8,8}
C){4,8,8}
D){4,8,13}
E)All distinct resamples are equally likely.

Question 17

Consider the following sample of four measurements {2,5,11,14}.Five bootstrap samples of this sample produced the following resamples: {2,5,5,14},{5,11,11,11},{2,11,14,14}, $\{ 2,2,14,14 \}$ , $\{ 5,5,11,14 \}$ .Based on these five resamples,find the bootstrap of the standard error of the sample median.

A)8.9
B)2.9240
C)2.3333
D)8.55
E)1.3077

Question 18

A random sample of 50 adult males in Toronto,Ontario were randomly selected and their weight,in kilograms,was recorded to the nearest kilogram.In order to compute a bootstrap confidence interval for the median,200 subsequent bootstrap samples were taken and the resulting sample medians were ordered from lowest to highest.The first 10 and last 10 ordered medians are given below: First 10: 68.0,68.5,69.0,69.0,69.5,70.0,70.0,70.0,70.5,70.5
Last 10: 93.5,94.0,94.0,94.5,94.5,95.0,95.0,95.5,95.5,96.0
Use the percentile method to compute a 95% bootstrap confidence interval for the true median weight of adult males in Toronto,Ontario.

A)(70.0,95.5)
B)(69.5,95.0)
C)(70.0,94.5)
D)(69.0,95.5)
E)(69.5,94.5)

Question 19

Consider the following sample of three measurements {4,8,13}.What is the probability of observing the distinct resample {4,8,13}?

A)2/9
B)1/3
C)1/27
D)0
E)1

Question 20

Consider the following sample of four measurements {1,2,3,4}.How many distinct resamples are possible?

A)4
B)10
C)24
D)35
E)27

Question 21

A random sample of 100 adult men in Toronto was randomly selected.The sample correlation between their heights and weights was found to be 0.62.Ten subsequent bootstrap samples based on this initial sample produced the following correlation coefficients: 0.58,0.52,0.67,0.62,0.63,0.54,0.71,0.66,0.58,0.70.Based on these ten resamples,find the bootstrap of the standard error of the sample correlation.

A)0.0206
B)0.2222
C)0.6210
D)0.0043
E)0.0652

Question 22

A grocery store manager is interested in determining whether or not a difference exists between the shelf life of two different brands of doughnuts.A random sample of 100 boxes of each brand was selected and the shelf life in days was determined for each box.The difference in sample means is found to be $\bar { x } _ { 1 } - \bar { x } _ { 2 } = 0.8$ .Ten subsequent bootstrap samples from each of these original samples yield the following values of $\bar { x } _ { 1 } ^ { * } - \bar { x } _ { 2 } ^ { * }$ 1.8,2.7,-1.9,2.1,0.1,-0.5,-2.3,1.2,2.1,-1.0.Based on these ten values,find the bootstrap of the standard error of $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ .

A)1.7982
B)0.5686
C)3.2334
D)0.43
E)2.1111

Question 23

A random sample of 50 students was randomly selected from your campus.Each student was asked to record their income and their average grade,as a percentage,for the 2010/2011 school year.The sample correlation coefficient was found to be -0.43.Ten subsequent bootstrap samples based on this initial sample produced the following correlation coefficients: -0.51,-0.33,-0.42,-0.45,-0.48,-0.44,-0.48,-0.56,-0.38,-0.39.Based on these ten resamples,find the bootstrap of the standard error of the sample correlation.

A)0.0214
B)0.0005
C)-0.444
D)0.2135
E)0.0675

Question 24

When selecting a bootstrap sample,which of the following statements is true?

A)We select another sample from the population of interest.This resample must have the same number of observations as the original sample.
B)We select a sample with replacement from the original sample observations.This resample must have the same number of observations as the original sample.
C)We select another sample from the population of interest.This resample must have more observations than the original sample in order to increase the accuracy of our estimates.
D)We select a sample without replacement from the original sample observations.This resample must have the same number of observations as the original sample.
E)We select a sample with replacement from the original sample observations.This resample must have fewer observations than the original sample.

Question 25

When computing a bootstrap confidence interval using the percentile method,what is the minimum number of resamples required (according to the textbook)?

A)200
B)1000
C)30
D)60
E)100

Question 26

Two brands of flares - Brand 1 and Brand 2 - are tested for their burning times (in minutes).A random sample of 40 of each brand of flare are burned and timed,and the difference in sample means is found to be $\bar { x } _ { 1 } - \bar { x } _ { 2 } = 4.3$ .Ten subsequent bootstrap samples from each of these original samples yield the following values of $\bar { x } _ { 1 } ^ { * } - \bar { x } _ { 2 } ^ { * }$ : 5.8,1.7,2.9,4.9,6.1,5.2,7.0,5.5,4.6,3.3.Based on these ten values,find the bootstrap of the standard error of $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ .

A)1.6193
B)0.5121
C)4.7
D)2.6222
E)3.4545

Question 27

When using bootstrapping to estimate the standard error of a statistic,what is the minimum number of resamples required (according to the textbook)?

A)1000
B)60
C)30
D)100
E)200