Question 1

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

Accepted Answer

The bias is given by the difference between the mean of the resamples and the observed difference in sample mean incomes. Therefore, we have: 
Bias = $7393 - $6987 = $406 
Therefore, the correct answer is B.

Question 2

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

Accepted Answer

The answer of The Q-Q plot below shows the bootstrap...

Question 3

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

Accepted Answer

The scenario involves selecting samples with replacement from the original data set, which is the key characteristic of the bootstrap resampling method. Therefore, the correct choice is A. Permutation involves rearranging the observed data points, which is not applicable in this scenario.

Question 4

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

Accepted Answer

The presence of a large bias suggests that the data may not be normally distributed, making the Bootstrap t interval less reliable. Additionally, the Q-Q plot suggests that the distribution of the bootstrap sample may not match the population distribution, making the Bootstrap percentile interval less reliable. The best option would be to use the bias-corrected accelerated (BCa) interval, which adjusts for bias and skewness in the bootstrap distribution.

Question 5

Bootstrapping can be performed on the ______.
A)mean
B)median
C)25% trimmed mean
D)All of the above
E)Only A and C

Accepted Answer

Bootstrapping is a resampling technique that can be applied to various statistics, including the mean, median, and trimmed means, to estimate the distribution of these statistics.

Question 6

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.

Accepted Answer

Bootstrap distribution based on small samples will generate bootstrap confidence intervals that are either too small or too long, depending on the specific sample and the method used for generating the intervals. This is because with small samples, the bootstrap resamples may not fully capture the variability of the population, leading to biased or inefficient estimates of the sampling distribution. Therefore, bootstrap intervals based on small samples should be interpreted with caution and preferably complemented with other inferential methods.

Question 7

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.

Accepted Answer

The BCa (Bias-Corrected and accelerated) interval is typically the best choice for bootstrap intervals as it corrects for bias and adjusts for skewness in the bootstrap distribution. It also takes into account asymmetry in the distribution of the bootstrap samples. Therefore, it is more accurate than the other intervals, especially when the sample size is smaller.

Question 8

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

Accepted Answer

Both statements are true because a small bias in the bootstrap distribution and a distribution close to Normal both contribute to the agreement between the bootstrap t and percentile confidence intervals.

Question 9

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

Accepted Answer

The distribution of the difference in means is based on resamples taken without replacement, which is a characteristic of permutation resampling. Therefore, the type of resampling method used in this scenario is permutation.

Question 10

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.

Accepted Answer

When using a permutation test with a large number of resamples (such as 1600), the estimated P-value can be considered approximately normally distributed. Therefore, we can use the formula for the standard deviation of a binomial proportion to estimate the standard deviation of the P-value: sqrt(p(1-p)/n), where p is the estimated P-value (0.02) and n is the number of resamples (1600). Plugging in these values, we get sqrt(0.02*0.98/1600) = 0.0035.

Question 11

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.

Accepted Answer

In a permutation test, the null hypothesis is that the two populations are identical. Thus, the test is only valid under this condition. Choice A is incorrect because the populations may have different means under the null hypothesis. Choice C is incorrect because the standard deviation can vary between the two populations. Choice D is incorrect because permutation tests do make assumptions about the null hypothesis.

Question 12

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.

Accepted Answer

Permutation tests do not assume the null hypothesis is true for constructing confidence intervals; they are used to test hypotheses. Confidence intervals are typically constructed using other methods, such as bootstrap resampling.

Question 13

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

Accepted Answer

The bootstrap distribution is based on resampling from the original sample. If the original sample size is very small, then resampling can only generate a limited number of unique samples. Therefore, the bootstrap distribution will not reflect the true variation in the population and may not be reliable.

Question 14

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

Accepted Answer

The answer of In which of the following situations can...

Question 15

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.

Accepted Answer

The bootstrap BCa interval is the best choice because it takes into account both the skewness and bias of the distribution, which is evident in the graph. The bootstrap t interval assumes normality, which is not a safe assumption in this case, and the bootstrap percentile interval does not take into account bias. Therefore, the BCa interval provides a more accurate estimate of the difference in means.

Question 16

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)(-$\infty$,0.391)

Accepted Answer

The 95% BCa confidence interval is given by the values in the 2.5% and 97.5% columns. Therefore, the 95% BCa confidence interval is (0.0253, 0.431).

Question 17

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.

Accepted Answer

Permutation test does not require any assumption regarding the distribution of the population, whereas a two-sample t-test requires the assumption that the population is normally distributed. Therefore, the permutation test is more reliable than the t-test when the population distribution is not known or not normal. The permutation test provides accurate P-values regardless of the distribution of the data.

Question 18

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)$

Accepted Answer

The answer of A consumer advocate is comparing income for...

Question 19

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

Accepted Answer

The P-value in a permutation test is calculated by locating the value of the test statistic on the permutation distribution and determining the proportion of permutations that are as extreme or more extreme than the observed test statistic. This approach does not rely on standard critical values or bootstrap estimates, but directly on the distribution of the test statistic under the permutation samples.

Question 20

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

Accepted Answer

Bootstrap distributions are created by resampling from the original sample. As such, the variation is largely based on the selection of the sample from the population. Different samples will result in different bootstrap distributions. The distribution of the population may influence the initial sample, but once the sample is selected, the bootstrap variation is solely based on resampling from that sample. The mean of the sample is also important in calculating the bootstrap distribution, but it is not the primary factor in determining the variation.

Quiz 16: Bootstrap Methods and Permutation Tests

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?

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?

Bootstrapping can be performed on the ______.

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

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?

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?

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

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

Is the bootstrap distribution from a very small sample reliable?

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

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

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

Variation in bootstrap distributions is largely based on _______.