Exam 30: The Bootstrap

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 mean.

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 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 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 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.

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

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).

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

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

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 $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ .

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

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.

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

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.

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

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

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

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.

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.

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.