Deck 16: Bootstrap Methods and Permutation Tests

Question 1

A simple random sample of size four is selected from a certain population.The sample yields the following data. 1 8 4 3
Suppose we bootstrap the mean of these data by drawing 1000 resamples.Suppose we find that the mean of the bootstrap distribution is 4.01.What is the bootstrap estimate of the bias of the statistic?

A)0.01
B)The difference between 4.01 and the unknown population mean
C)This cannot be calculated without knowing the actual values of each of the 1000 resamples.
D)Zero,since the sample mean is an unbiased estimator of the population mean

Answer 1

0.01

Question 2

A simple random sample of corporate executives is taken and their annual salaries recorded.The goal is to estimate the 25% trimmed mean of the annual salaries of the population of all corporate executives.To do this,we bootstrap the 25% trimmed mean using our sample and obtain the bootstrap distribution. Suppose we sampled 1500 corporate executives.Summary statistics are given below. What is the bootstrap,t,95% confidence interval for the 25% population trimmed mean?

A)$237,448.22
$8,068.40
B)$237,601.97
$8,068.40
C)$237,601.97
$301.35
D)A t interval is probably not safe to use in this scenario.

Answer 2

$237,601.97
$8,068.40

Question 3

A simple random sample of size four is selected from a certain population.The sample yields the following data. 1 8 4 3
Suppose we bootstrap the mean of these data by drawing 1000 resamples.Which of the following could not be the means of the first three resamples?

A)1,4,8
B)2.25,3.25,4.25
C)4,5,6
D)4.5,5.5,6.5

Answer 3

4.5,5.5,6.5

Question 4

Which of the following statements about bootstrap estimates is/are TRUE?

A)The bootstrap is able to estimate the sampling distribution,bias,and standard error of a wide variety of statistics.
B)A small bootstrap estimate of bias suggests that the sampling distribution of the statistic is centered at the population parameter.
C)The bootstrap estimate of bias is determined from the mean of the bootstrap distribution and the value of the statistic in the original sample.
D)If the bootstrap distribution of a statistic is approximately Normal and the bias is small,a bootstrap confidence interval for the population parameter can be constructed of the form statistic ± t*SE_boot.
E)All of the above are true.

Question 5

A random sample of size 25 was selected from a population and from that sample the value of a statistic,which we will label ,was determined to be = 6.250.Using the bootstrap method,5000 resamples were obtained from the sample observations.The histogram of the values of the statistic from these 5000 resamples looked reasonably Normal.The 5000 's had a mean of 6.234 and standard deviation of 1.493.Which of the following statements about this situation is/are TRUE?

A)For the statistic ,the bootstrap standard error is
= 0.2986.
B)The bootstrap estimate of bias in the statistic is 0.016.
C)The bootstrap standard error is 1.493.
D)None of the above is true.
E)Only B and C are true.

Question 6

Residents of large urban cities often complain about the lack of parking and overpriced parking tickets.The mayor of Washington,DC,is doing a study to find out the 25% trimmed mean of all parking tickets received by residents in 1 year.A sample of 50 residents yielded a 25% trimmed mean of $150.00 and the bootstrap standard error is $23.00.What is the distribution of the critical value used to computer a 95% confidence interval?

A)t(25)
B)t(49)
C)N(0,1)
D)None of the above

Question 7

A simple random sample of corporate executives is taken and their annual salaries recorded.The goal is to estimate the 25% trimmed mean of the annual salaries of the population of all corporate executives.To do this,we bootstrap the 25% trimmed mean using our sample and obtain the bootstrap distribution. Suppose we sampled 15 corporate executives.Summary statistics are given below. What is the bootstrap,t,95% confidence interval for the 25% population trimmed mean?

A)$277,448.22
$82,944.08
B)$287,601.97
$82,944.08
C)$287,601.97
$90,180.53
D)A t interval is probably not safe to use in this scenario.

Question 8

Give an example of the plug-in principle.

A)Estimate the population mean by the sample mean.
B)Estimate the population mean by the sample median.
C)Estimate the sample mean by the population mean.
D)None of the above

Question 9

The general procedure for bootstrapping calls for resamples from the data ______.

A)with replacement
B)without replacement
C)with or without placement

Question 10

Which of the following statements about the bootstrapping method is/are FALSE?

A)The bootstrap distribution is created by resampling with replacement from the sample that was selected from the population.
B)The bootstrap distribution of a statistic,based on the resamples,represents the sampling distribution of the statistic.
C)The bootstrap standard error,SE_boot,of a statistic is the standard deviation of the bootstrap distribution of the statistic.
D)The bootstrap distribution is centered close to the mean of the population from which the original sample is obtained.
E)When resampling to form the bootstrap distribution,each of the resamples must be of the same size as the original sample from the population.

Question 11

Bootstrapping is useful because it adds more values to the original data set.

A)True.You are able to add as many new values as needed to the original data set.
B)True.You are able to add a finite amount of new values to the original data set.
C)False.No new values are added to the data set.
D)False.Bootstrapping is only used to change the mean of the original data set.

Question 12

Suppose we are interested in the amount of time (in minutes)college students spend on Facebook per day.A simple random sample of size five is taken and we obtain the numbers 120,25,365,22,and 154.Which of the following is a possible resample for bootstrapping?

A)25,25,25,25,120
B)120,25,1,22,154
C)25,25,25,25,24
D)None of the above are possible resamples.

Question 13

How is the bootstrap standard error calculated?

A)It is the difference between the estimate of a parameter obtained by bootstrapping and the estimate obtained using classical methods.
B)It is the standard deviation divided by the square root of the sample size.
C)It is the standard deviation of the bootstrap distribution.
D)It is the standard deviation of the original sample that is used to bootstrap the statistic.

Question 14

Suppose you are performing bootstrapping using computer software.The software output gives the mean of the original data set to be 15.The software output gives the bias to be 2.What is the mean of the resample means?

A)17
B)13
C)2
D)This cannot be determined from the given information.

Question 15

Residents of large urban cities often complain about the lack of parking and overpriced parking tickets.The mayor of Washington,DC,is doing a study to find out the 25% trimmed mean of all parking tickets received by residents in 1 year.A sample of 50 residents yielded a 25% trimmed mean of $150.00 and the bootstrap standard error is $23.00.What is the 95% bootstrap confidence interval for the 25% trimmed mean?

A)(104,196)
B)(102.62,197.38)
C)This cannot be computed from the given information.
D)None of the above

Question 16

A random sample of size 25 was selected from a population and from that sample the value of a statistic,which we will label ,was determined to be = 6.250.Using the bootstrap method,5000 resamples were obtained from the sample observations.The histogram of the values of the statistic from these 5000 resamples looked reasonably Normal.The 5000 's had a mean of 6.234 and standard deviation of 1.493.An approximate 95% confidence interval for the parameter that corresponds to is given by

A)6.234 ± 2.064(0.299).
B)6.234 ± 2.064(1.493).
C)6.250 ± 2.064(1.493).
D)6.250 ± 2.064(0.299).
E)6.250 ± 1.960(1.493).

Question 17

Which of the following statements is TRUE regarding inference based on a small sample?

A)Traditional methods of inference based on a small sample are often unreliable.Bootstrap methods overcome this problem.
B)Bootstrap methods based on a small sample are unreliable.One should use traditional methods with small samples.
C)Any inference based on a small sample,including bootstrap inference,should be treated with caution.
D)Increasing the number of resamples is necessary to guarantee that inference based on a small sample is trustworthy.

Question 18

Residents of large urban cities often complain about the lack of parking and overpriced parking tickets.The mayor of Washington,DC,is doing a study to find out the 25% trimmed mean of all parking tickets received by residents in 1 year.A sample of 50 residents yielded a 25% trimmed mean of $150.00 and the bootstrap standard error is $23.00.A 95% bootstrap confidence interval will be accurate only if what condition is met?

A)The bias is small.
B)The bootstrap distribution is approximately Normal.
C)The bias is large.
D)A and B are both correct answers.

Question 19

The bootstrap distribution for means calculated from a large number of resamples should be ______.

A)Normal
B)T
C)binomial
D)No distribution assumption is required.

Question 20

The following random sample was obtained from a population with unknown mean and standard deviation. From the sample the following statistics were calculated: = 10 and s = 3.16. This sample was used to form the bootstrap distribution of the sample standard deviation based on 1000 resamples.Which of the following statements with respect to this bootstrapping process and distribution is/are FALSE?

A)The distribution of the 1000 resample standard deviations represents the sampling distribution of s.
B)One of the many possible resamples is { 10,12,8,8,6},for which the resample mean and standard deviation are = 8.8 and
= 2.28.
C)The bootstrap mean for the statistic is given by .
D)The bootstrap standard error of the statistic of interest is given by

E)The bootstrap estimate of bias is the difference between the mean of the bootstrap distribution of the 1000 s*'s and the sample standard deviation,s = 3.16.

Question 21

A consumer advocate is comparing income for a wide variety of jobs between the South and the Northeast.The average income of a simple random sample of 800 jobs in the Northeast is compared to the average income of a simple random sample of 750 jobs in the South.From the collected data,950 resamples are taken.Each time,750 incomes are selected with replacement from the data collected from the South,and 800 incomes are selected with replacement from the data collected from the Northeast.The distribution of the difference in means based on the resamples is shown below (in thousands of dollars). The mean of the resamples is $7393,and the standard deviation of the resamples is $10,807.The observed difference in sample mean incomes was $6987.What is the bootstrap estimate of the bias?

A)$200
B)$406
C)$3414
D)$3820

Question 22

The Q-Q plot below shows the bootstrap distribution of a trimmed mean with a small bias and a large original sample.What method should be used to generate a 95% bootstrap confidence interval?

A)Bootstrap t
B)Bootstrap percentile
C)Bootstrap bias-corrected accelerated (BCa)interval
D)A or B

Question 23

A consumer advocate is comparing income for a wide variety of jobs between the South and the Northeast.The average income of a simple random sample of 800 jobs in the Northeast is compared to the average income of a simple random sample of 750 jobs in the South.From the collected data,950 resamples are taken.Each time,750 incomes are selected with replacement from the data collected from the South,and 800 incomes are selected with replacement from the data collected from the Northeast.The distribution of the difference in means based on the resamples is shown below (in thousands of dollars). What type of resampling method was used in this scenario?

A)Bootstrap
B)Permutation
C)Neither bootstrap nor permutation

Question 24

The Q-Q plot below shows the bootstrap distribution of a trimmed mean with a large bias and a large original sample.What method should be used to generate a 95% bootstrap confidence interval?

A)Bootstrap t
B)Bootstrap percentile
C)Bootstrap bias-corrected accelerated (BCa)interval
D)A or B

Question 25

Bootstrapping can be performed on the ______.

A)mean
B)median
C)25% trimmed mean
D)All of the above
E)Only A and C

Question 26

Bootstrap distribution based on small samples will generate bootstrap confidence intervals that are _____________.

A)too small
B)too long
C)accurate
D)A and B are correct.

Question 27

A large simple random sample is selected from a certain population and the statistic of interest is calculated.Suppose we bootstrap this statistic and obtain the following bootstrap distribution. Which of the following 95% bootstrap intervals is best to use?

A)The bootstrap t interval
B)The bootstrap percentile interval
C)The BCa interval
D)Because the sample was large,we can use any of the above.

Question 28

Consider the following two statements. I: The bootstrap t confidence interval and bootstrap percentile confidence interval will agree closely when the bias of the bootstrap distribution is small.
II: The bootstrap t confidence interval and bootstrap percentile confidence interval will agree closely when the bootstrap distribution is close to Normal.
Which of these statements is TRUE?

A)I
B)II
C)Neither I nor II
D)Both I and II

Question 29

A real estate agency with property all throughout the Midwest is interested in home prices within the state of Michigan.The average price of a simple random sample of residential properties in the Upper Peninsula is compared to the average price of a simple random sample of residential properties in the Lower Peninsula.From the collected data,2440 resamples without replacement are taken.The distribution of the difference in means based on the resamples is shown below (in thousands of dollars). What type of resampling method was used in this scenario?

A)Bootstrap
B)Permutation
C)Neither bootstrap nor permutation

Question 30

Suppose we use a permutation test to test a one-sided alternative hypothesis and estimate the P-value to be 0.02.If we used 1600 resamples,what is the approximate standard deviation of the estimated P-value?

A)0.0000123
B)0.0005
C)0.0035
D)This question makes no sense.P-values do not have standard deviations.

Question 31

Under what condition is a permutation test for comparing two populations valid?

A)The populations have the same mean under the null hypothesis.
B)The populations are identical under the null hypothesis.
C)The two populations have the same standard deviation under both the null and alternative hypotheses.
D)Under all circumstances-the permutation test does not require any particular assumptions and hence is very powerful.

Question 32

Which of the following statements about permutation resamples and tests is/are FALSE?

A)Permutation resamples are drawn without replacement,unlike bootstrap resamples,which are drawn with replacement.
B)Permutation resamples must be drawn in a way that is consistent with the null hypothesis and with the study design.
C)Permutation tests do not require specific population shapes and,regardless of the shape and size of the population,give accurate P-values.
D)Accurate permutation confidence intervals can be constructed assuming that the null hypothesis is true.
E)The P-value of the permutation test is found by locating the original value of the statistic on the permutation distribution.The proportion of resamples with statistics that exceed the original observed statistic is the P-value.

Question 33

Is the bootstrap distribution from a very small sample reliable?

A)Yes,because the sample size does not matter
B)No,because the distribution will not likely match the population
C)None of the above

Question 34

In which of the following situations can we use a permutation test?

A)Two-sample problems when the null hypothesis says the two populations are identical
B)Matched-pairs designs when the null hypothesis says that the population medians are equal
C)Problems in which we examine relationships between two qualitative variables
D)All of the above

Question 35

A consumer advocate is comparing income for a wide variety of jobs between the South and the Northeast.The average income of a simple random sample of 800 jobs in the Northeast is compared to the average income of a simple random sample of 750 jobs in the South.From the collected data,950 resamples are taken.Each time,750 incomes are selected with replacement from the data collected from the South,and 800 incomes are selected with replacement from the data collected from the Northeast.The distribution of the difference in means based on the resamples is shown below (in thousands of dollars). Based on the graph,which of the following 95% bootstrap intervals is best to use to estimate the difference in mean income?

A)The bootstrap t interval
B)The bootstrap percentile interval
C)The BCa interval
D)Because the samples were large,we can use any of the above.

Question 36

A large simple random sample is selected from a certain population and the statistic of interest is calculated.Suppose we bootstrap this statistic and wish to compute the 95% BCa confidence interval.We obtain the following computer output. BCa Confidence Intervals:
2.5% 5% 95% 97.5%
Statistic 0.0253 0.0568 0.391 0.431
What is the 95% BCa confidence interval?

A)(0.0253,0.431)
B)(0.0568,0.391)
C)(0,0.391)
D)(-,0.391)

Question 37

What is the main advantage of using a permutation test over a two-sample t test?

A)The permutation test can be conducted more quickly than a two-sample t test and does not require the use of specialized tables.
B)The permutation test is valid for much smaller sample sizes.
C)The permutation test gives exact P-values.The two-sample t test gives only approximate P-values.
D)The permutation test gives accurate P-values even when the sampling distribution of the difference in means is not close to Normal.

Question 38

A consumer advocate is comparing income for a wide variety of jobs between the South and the Northeast.The average income of a simple random sample of 800 jobs in the Northeast is compared to the average income of a simple random sample of 750 jobs in the South.From the collected data,950 resamples are taken.Each time,750 incomes are selected with replacement from the data collected from the South,and 800 incomes are selected with replacement from the data collected from the Northeast.The distribution of the difference in means based on the resamples is shown below (in thousands of dollars). The mean of the resamples is $7393,and the standard deviation of the resamples is $10,807.What is the value of the bootstrap standard error?

A)$10,807
B)$
C)$
D)$

Question 39

How do we calculate the P-value for a permutation test?

A)By comparing the bootstrap estimate to critical values in standard statistical tables
B)By locating the value of the test statistic on the permutation distribution
C)By computing the percentage of times the permutation resamples lead to values of the test statistic that exceed the standard critical value
D)By determining whether the value under the null hypothesis is outside the permutation confidence interval

Question 40

Variation in bootstrap distributions is largely based on _______.

A)the distribution of the population
B)the selection of the sample from the population
C)the mean of the sample
D)None of the above

Question 41

The mayor of Beltsville has commissioned a research company to estimate the average age of the houses in a large historical district in the town.Data on 120 randomly selected properties are collected,and a histogram of the sample data shows a right-skewed distribution with several large outliers.The person leading this project decides that the standard t procedures should not be used.He decides to use the 25% trimmed mean as a measure of center.The leader of the project then decides to bootstrap the 25% trimmed mean.The bootstrap distribution is shown below.Some summary measures are given as well. Number of Replications: 500
Summary Statistics:
Observed Mean Bias SE
TrimMean 53.675 53.05 4.67
Percentiles:
2.5% 5% 25% 50% 75% 95% 97.5%
TrimMean 41.05 44.28 50.45 53.20 56.05 60.80 63.82
The dashed line in the histogram represents the mean of the resamples.The solid line represents the observed value of the trimmed mean in the original sample.
As part of the summary measure,the value 4.67 is reported as the SE.How was this value calculated,and what does it represent?

Question 42

A real estate agency with property all throughout the Midwest is interested in home prices within the state of Michigan.The average price of a simple random sample of residential properties in the Upper Peninsula is compared to the average price of a simple random sample of residential properties in the Lower Peninsula.From the collected data,2440 resamples without replacement are taken.The distribution of the difference in means based on the resamples is shown below (in thousands of dollars). The observed difference in sample mean prices is $300,000.What is the estimate of the P-value?

A)3/2440
B)3/2439
C)4/2440
D)4/2441