College Statistics Quiz: Bootstrap Resampling
20 questions · exam conditions
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Bootstrap ResamplingQuestion 1 of 20

An original sample consists of the five values {2, 4, 4, 7, 9}. One possible bootstrap sample is {9, 4, 2, 4, 9}. What is the relationship between the original sample's median and the bootstrap sample's median?

The bootstrap sample median (4) is one of many bootstrap statistics that would be used to form a distribution centered at the original sample median (4).
The bootstrap sample median must always be equal to the original sample median.
The bootstrap sample median is guaranteed to be a better estimate of the population median than the original sample median.
The process is invalid because the value '4' from the original sample was drawn twice.
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College Statistics Quiz

College Statistics Quiz: Bootstrap Resampling

Practice Bootstrap Resampling in College Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Bootstrap Resampling, giving you a quick way to practice the rules, question types, and explanations that matter most for College Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

An original sample consists of the five values {2, 4, 4, 7, 9}. One possible bootstrap sample is {9, 4, 2, 4, 9}. What is the relationship between the original sample's median and the bootstrap sample's median?

  1. The bootstrap sample median (4) is one of many bootstrap statistics that would be used to form a distribution centered at the original sample median (4). (correct answer)
  2. The bootstrap sample median must always be equal to the original sample median.
  3. The bootstrap sample median is guaranteed to be a better estimate of the population median than the original sample median.
  4. The process is invalid because the value '4' from the original sample was drawn twice.
Explanation: The median of the original sample {2, 4, 4, 7, 9} is 4. The median of the bootstrap sample {2, 4, 4, 9, 9} (sorted from {9, 4, 2, 4, 9}) is also 4. This specific value is one 'bootstrap statistic'. The full bootstrap procedure involves generating many such bootstrap samples, calculating the median for each one, and collecting these medians into a bootstrap distribution. This distribution is then used to infer properties (like standard error or a confidence interval) of the median. The distribution of these bootstrap medians will be centered around the original sample median of 4.

Question 2

A study measures the weight of 20 subjects before and after a diet program. To construct a bootstrap confidence interval for the mean difference in weight (after - before), what is the correct unit to resample?

  1. The 20 'before' weights and the 20 'after' weights are resampled independently of each other.
  2. All 40 weight measurements are pooled together into one large group, and 40 values are drawn from it.
  3. The 20 pairs of (before, after) weights, ensuring that each subject's pair of measurements is kept together during resampling. (correct answer)
  4. The 20 'before' weights are resampled, and the 20 'after' weights are resampled, and then the results are paired randomly.
Explanation: This is a paired data problem. The crucial information is the link between each subject's 'before' and 'after' measurement. To preserve this structure, the resampling must be done on the unit of observation, which is the individual subject. By resampling the pairs of (before, after) weights, we maintain the connection. After resampling, one would calculate the difference for each pair and then find the mean of those differences. Resampling the 'before' and 'after' weights independently (A or D) or pooling them (B) would break this pairing and would be appropriate only for a two-independent-samples test, which is incorrect for this study design.

Question 3

A researcher computes a 95% bootstrap confidence interval for a population mean and a 99% bootstrap confidence interval from the same bootstrap distribution. How will the 99% interval compare to the 95% interval?

  1. The 99% interval will be wider than the 95% interval. (correct answer)
  2. The 99% interval will be narrower than the 95% interval.
  3. Both intervals will have the same width, but the 99% interval will have a different center.
  4. It is impossible to predict the relationship without knowing the shape of the bootstrap distribution.
Explanation: A confidence interval must capture a certain percentage of the most plausible values for a parameter. To have a higher level of confidence (e.g., 99% vs. 95%), one must include a wider range of values to be more certain that the true parameter is contained within the interval. For a percentile bootstrap interval, the 95% interval spans from the 2.5th to the 97.5th percentiles of the distribution, while the 99% interval spans from the 0.5th to the 99.5th percentiles. The latter is necessarily a wider range.

Question 4

A researcher is studying the 90th percentile of household income from a sample of data that is known to be severely right-skewed. Traditional parametric methods for creating a confidence interval for a percentile are complex and their assumptions are not met. Why is bootstrap resampling a particularly appropriate method in this scenario?

  1. It transforms the skewed sample data into a normal distribution, allowing for the use of standard formulas.
  2. It provides an empirical estimate of the sampling distribution without making strong assumptions about the population's distribution. (correct answer)
  3. It is guaranteed to produce a narrower and more precise confidence interval than any traditional method.
  4. It finds the true population 90th percentile by using the sample data as a proxy for the population.
Explanation: The primary advantage of bootstrapping is that it is a non-parametric method. It does not require assumptions about the underlying population distribution (e.g., normality). This makes it ideal for skewed data and for complex statistics like percentiles where simple formulas for the sampling distribution do not exist. Choice A is incorrect; bootstrapping does not transform the original data. Choice C is incorrect; the width of the interval reflects the uncertainty and is not guaranteed to be narrower. Choice D is incorrect; bootstrapping provides a confidence interval, which is an estimate of a range for the parameter, not the exact true value.

Question 5

A statistician generates 2,000 bootstrap resamples to estimate the standard error of the interquartile range (IQR) of a dataset. They compute the IQR for each of the 2,000 bootstrap samples, creating a bootstrap distribution of IQRs. How should they calculate the bootstrap estimate of the standard error of the IQR?

  1. Calculate the standard deviation of the 2,000 bootstrap IQR values. (correct answer)
  2. Calculate the standard deviation of the original sample and divide by n\sqrt{n}.
  3. Find the range of the 2,000 bootstrap IQR values and divide by 4.
  4. Calculate the IQR of the 2,000 bootstrap IQR values.
Explanation: The bootstrap estimate of the standard error of a statistic is the standard deviation of the bootstrap distribution of that statistic. In this case, the statistic of interest is the IQR. Therefore, the standard error is estimated by taking the standard deviation of the 2,000 calculated bootstrap IQRs. Choice B is the formula for the standard error of the mean, which is not applicable to the IQR. Choices C and D describe other measures of spread for the bootstrap distribution, but they are not the definition of the bootstrap standard error.

Question 6

A 95% bootstrap confidence interval for the difference in mean test scores between two independent groups (Experimental - Control) is found to be [2.1, 10.8]. Based on this interval, what is the most appropriate conclusion at a 5% significance level?

  1. There is a 95% probability that the true mean difference lies between 2.1 and 10.8 points.
  2. The mean score for the Experimental group is higher, but the difference is not statistically significant.
  3. There is statistically significant evidence that the mean score is higher for the Experimental group than the Control group. (correct answer)
  4. The interval is too wide to make a conclusion, so the study should be repeated with a larger sample.
Explanation: The confidence interval provides a range of plausible values for the true difference in population means. Since the entire interval [2.1, 10.8] is above zero, the value of zero (indicating no difference) is not a plausible value. This is equivalent to rejecting the null hypothesis of no difference at the α=0.05\alpha = 0.05 significance level. Therefore, there is statistically significant evidence of a positive difference, meaning the Experimental group's mean score is higher. Choice A is a common misinterpretation of confidence intervals. Choice B is contradictory; since zero is not in the interval, the result is statistically significant.

Question 7

A bootstrap procedure is used to find a 95% confidence interval for the population median. The original sample is {5, 8, 12, 25, 30}. The sample median is 12. If one of the bootstrap samples is {5, 5, 8, 12, 12}, what is the bootstrap statistic calculated from this particular sample?

  1. 5, which is the minimum value of this bootstrap sample.
  2. 12, which is the median of the original sample.
  3. 8.4, which is the mean of this bootstrap sample.
  4. 8, which is the median of this bootstrap sample. (correct answer)
Explanation: When you encounter bootstrap questions, remember that the bootstrap statistic is the same type of statistic as your original parameter of interest, calculated from each bootstrap sample. Since we're finding a confidence interval for the population median, each bootstrap sample must yield its own median value. For the bootstrap sample {5, 5, 8, 12, 12}, you need to find the median. With 5 values arranged in order, the median is the middle (3rd) value, which is 8. This bootstrap statistic of 8 would be one of many values used to construct the confidence interval for the population median. Looking at the wrong answers: Choice A (5) identifies the minimum value rather than the median of the bootstrap sample. Choice B (12) represents the median of the original sample, but bootstrap procedures require calculating the statistic fresh from each new bootstrap sample—you can't just reuse the original sample's value. Choice C (8.4) calculates the mean of the bootstrap sample 5+5+8+12+125=8.4\frac{5+5+8+12+12}{5} = 8.4, but since we're estimating the population median, we need the sample median, not the sample mean. The correct answer is D: the bootstrap statistic is 8, the median of this particular bootstrap sample. Study tip: In bootstrap problems, always match the statistic to the parameter. If you're bootstrapping for a population median, calculate the median from each bootstrap sample. If bootstrapping for a population mean, calculate the mean from each sample. The bootstrap statistic type must align with what you're trying to estimate.

Question 8

An analyst has a sample of 35 data points that is strongly left-skewed. They compute a 95% confidence interval for the population mean using both a traditional t-interval and a bootstrap percentile interval. Which of the following is the most likely outcome?

  1. Both intervals will be identical because they are based on the same sample data and confidence level.
  2. The bootstrap interval will likely be asymmetric, with the sample mean closer to the upper bound, while the t-interval is symmetric. (correct answer)
  3. The t-interval is more trustworthy because the sample size is greater than 30, which guarantees its assumptions are met.
  4. The bootstrap interval is invalid for skewed data; the data must first be transformed to be symmetric.
Explanation: A t-interval is always symmetric around the sample mean (xˉ±t∗⋅SE\bar{x} \pm t^* \cdot SE). However, when the underlying data is strongly skewed, the sampling distribution of the mean will also be skewed, especially with a modest sample size like n=35n=35. The bootstrap percentile interval will capture this asymmetry. For left-skewed data, the left tail of the sampling distribution is longer, so the distance from the sample mean to the lower bound of the confidence interval will be greater than the distance to the upper bound. This means the sample mean will be closer to the upper bound.

Question 9

A researcher has an original sample of 30 measurements of stream acidity. To estimate the sampling distribution of the sample median, they decide to use bootstrap resampling. Which of the following correctly describes the process of generating a single bootstrap sample?

  1. Randomly select 30 measurements from the original sample without replacement.
  2. Randomly select 30 measurements from the original sample with replacement. (correct answer)
  3. Fit a theoretical distribution, such as a normal distribution, to the original sample and then draw 30 measurements from it.
  4. Calculate the median of the original sample and add random noise drawn from a standard normal distribution to create 30 new values.
Explanation: The core principle of bootstrap resampling is to simulate the process of drawing samples from a population by instead drawing samples from the original sample itself. This process must be done with replacement to allow for variability in the bootstrap samples. The bootstrap sample size should be the same as the original sample size. Choice B correctly describes this procedure. Choice A describes a permutation, not a bootstrap sample. Choices C and D describe other simulation methods (parametric bootstrap and general Monte Carlo methods, respectively), not the standard non-parametric bootstrap.

Question 10

A bootstrap distribution of 5,000 sample means is generated from an original sample of size n=40n=40. The mean of the original sample was 112.5. The resulting bootstrap distribution of means appears approximately normal. The center of this bootstrap distribution will be closest to which value?

  1. The mean of the population from which the original sample was drawn.
  2. The mean of the original sample, 112.5. (correct answer)
  3. 0, as the distribution is centered for standardization.
  4. The median of the original sample.
Explanation: A key property of the bootstrap distribution is that it is centered at the statistic from the original sample. Since resampling is done from the original sample, the average of the bootstrap sample statistics will converge to the original sample's statistic. It is an approximation of the sampling distribution centered at the sample statistic, not the (unknown) population parameter. Therefore, the center will be very close to the original sample mean of 112.5.

Question 11

After generating a bootstrap distribution of 4,000 sample medians, a student sorts these medians in ascending order. To construct an approximate 90% confidence interval for the population median using the percentile method, which values should they select from the sorted list?

  1. The value at the 40th position and the value at the 3960th position.
  2. The value at the 100th position and the value at the 3900th position.
  3. The value at the 200th position and the value at the 3800th position. (correct answer)
  4. The value at the 400th position and the value at the 3600th position.
Explanation: A 90% confidence interval leaves 10% of the distribution in the tails, with 5% in each tail. To find the cutoffs, we calculate 5% of the total number of bootstrap samples: 0.05×4000=2000.05 \times 4000 = 200. This means the lower bound of the interval is the 200th value in the sorted list, and the upper bound is the value that leaves 5% at the top, which is the 4000−200=38004000 - 200 = 3800th value. Choice A corresponds to a 99% interval. Choice B corresponds to a 95% interval. Choice D corresponds to an 80% interval.

Question 12

A student attempts to create a bootstrap distribution for the mean from a sample of size n=25n=25. They mistakenly generate 1,000 resamples of size 25 without replacement. What will be the standard deviation of the resulting distribution of the 1,000 sample means?

  1. Approximately equal to the standard deviation of the original sample, ss.
  2. Approximately equal to the standard error of the mean, s/25s/\sqrt{25}.
  3. It cannot be determined without knowing the values in the original sample.
  4. Exactly 0. (correct answer)
Explanation: When sampling without replacement from a sample of size nn to create a new sample of size nn, the new sample will always be a permutation of the original sample. It will contain the exact same set of numbers, just in a different order. The mean of a set of numbers does not depend on their order. Therefore, the mean of every single resample will be identical to the mean of the original sample. The standard deviation of a list of identical numbers is exactly 0.

Question 13

A political analyst creates two bootstrap distributions for a sample proportion. The first is based on B=1,000B=1,000 resamples and the second is based on B=20,000B=20,000 resamples from the same original data. How will the histogram for the second distribution most likely differ from the first?

  1. Its center will be significantly closer to the true population proportion.
  2. Its standard deviation will be substantially smaller, reflecting greater certainty.
  3. Its shape will be smoother and provide a more accurate approximation of the theoretical sampling distribution. (correct answer)
  4. Its overall shape will be the same, but it will be shifted to the right on the number line.
Explanation: The number of bootstrap resamples, BB, determines the resolution of the approximation to the sampling distribution. A larger BB does not change the center or the spread of the theoretical sampling distribution being estimated; it simply allows for a more detailed and stable picture of it. Therefore, the histogram with B=20,000B=20,000 will look like a smoother version of the one with B=1,000B=1,000. The center (determined by the original sample statistic) and standard deviation (determined by the original sample's variability and size nn) will not systematically change, only be estimated more precisely.

Question 14

An original sample consists of the five values {2, 4, 4, 7, 9}. One possible bootstrap sample is {9, 4, 2, 4, 9}. What is the relationship between the original sample's median and the bootstrap sample's median?

  1. The bootstrap sample median (4) is one of many bootstrap statistics that would be used to form a distribution centered at the original sample median (4). (correct answer)
  2. The bootstrap sample median must always be equal to the original sample median.
  3. The bootstrap sample median is guaranteed to be a better estimate of the population median than the original sample median.
  4. The process is invalid because the value '4' from the original sample was drawn twice.
Explanation: The median of the original sample {2, 4, 4, 7, 9} is 4. The median of the bootstrap sample {2, 4, 4, 9, 9} (sorted from {9, 4, 2, 4, 9}) is also 4. This specific value is one 'bootstrap statistic'. The full bootstrap procedure involves generating many such bootstrap samples, calculating the median for each one, and collecting these medians into a bootstrap distribution. This distribution is then used to infer properties (like standard error or a confidence interval) of the median. The distribution of these bootstrap medians will be centered around the original sample median of 4.

Question 15

A researcher collects a sample of size n1=40n_1=40. They generate a bootstrap distribution of the mean. A colleague collects a sample of size n2=160n_2=160 from the same population. They also generate a bootstrap distribution of the mean. Assuming both original samples happen to have very similar sample means, how will the second bootstrap distribution most likely compare to the first?

  1. It will have approximately the same width, as both estimate the variability of the same population parameter.
  2. It will be approximately twice as wide, because the sample size is four times larger.
  3. It will be approximately half as wide, because the standard error is inversely proportional to the square root of the sample size. (correct answer)
  4. Its center will be a more accurate estimate of the population mean, but its width will be unrelated to sample size.
Explanation: The bootstrap distribution's width is an estimate of the standard error of the statistic. The standard error of the mean is σ/n\sigma/\sqrt{n}. The bootstrap standard error will approximate s/ns/\sqrt{n}. Since the sample size nn increases from 40 to 160 (a factor of 4), the standard error will decrease by a factor of 4=2\sqrt{4}=2. Therefore, the bootstrap distribution generated from the larger sample will be approximately half as wide as the one from the smaller sample.

Question 16

A bootstrap distribution for the sample mode is created from a small original sample. The resulting histogram of bootstrap modes shows distinct, separate bars at only a few values. What is the most likely reason for this discrete, gappy appearance?

  1. The number of bootstrap resamples was too small, creating gaps where more samples would have filled in the values.
  2. The bootstrap procedure is not valid for discrete data or statistics like the mode and produces unreliable, gappy results.
  3. This indicates that the population is multimodal, with each bar representing a true population mode.
  4. The original sample had a limited number of unique values, so the mode of any bootstrap sample could only be one of those values. (correct answer)
Explanation: When you encounter bootstrap distribution questions, focus on understanding what the bootstrap procedure actually does: it creates new samples by resampling with replacement from your original data. The correct answer is D because bootstrap resampling can only draw values that already exist in the original sample. If your original sample contains only a few unique values, then any bootstrap sample created from it can only contain those same values. The mode of each bootstrap sample must therefore be one of the original values, creating distinct bars at only those specific points. This is a fundamental limitation of the bootstrap method - it cannot create new data values that weren't in the original sample. Option A incorrectly assumes that more bootstrap samples would fill in the gaps. However, no amount of additional resampling can produce values that don't exist in the original data. Option B is wrong because bootstrap procedures work perfectly well with discrete data and statistics like the mode - the discrete appearance doesn't indicate invalidity. Option C misinterprets what the bars represent. The bars show which values from the original sample appeared as modes in the bootstrap samples, not multiple population modes. Remember this key principle: bootstrap distributions are constrained by the original sample's unique values. When you see gappy, discrete bootstrap distributions, think about the diversity (or lack thereof) in the original data. This is especially important for statistics like the mode, median, or any order statistic when working with small samples containing few unique values.

Question 17

A researcher has a very small sample of n=8n=8 observations. They are concerned about the reliability of using bootstrap resampling with such a small initial sample. Why is this a valid concern?

  1. Bootstrap resampling always requires a sample size of at least 30 to produce a normal-shaped distribution.
  2. The bootstrap process cannot be performed if the sample size is less than 10 due to computational limitations.
  3. With a very small sample, the number of possible distinct bootstrap samples is limited, potentially creating a poor approximation of the sampling distribution. (correct answer)
  4. A small sample size forces the bootstrap confidence interval to be symmetric, which is often not appropriate.
Explanation: The effectiveness of the bootstrap depends on the original sample being a good representation of the population. A very small sample may not be representative and, critically, it contains very little information. The number of unique values is small, so the variety of possible bootstrap samples is limited. This can lead to a bootstrap distribution that is sparse, multimodal, and a poor reflection of the true sampling distribution's shape and spread. While bootstrap can be run on a small sample, the results should be interpreted with great caution.

Question 18

An analyst has a sample of 35 data points that is strongly left-skewed. They compute a 95% confidence interval for the population mean using both a traditional t-interval and a bootstrap percentile interval. Which of the following is the most likely outcome?

  1. Both intervals will be identical because they are based on the same sample data and confidence level.
  2. The bootstrap interval will likely be asymmetric, with the sample mean closer to the upper bound, while the t-interval is symmetric. (correct answer)
  3. The t-interval is more trustworthy because the sample size is greater than 30, which guarantees its assumptions are met.
  4. The bootstrap interval is invalid for skewed data; the data must first be transformed to be symmetric.
Explanation: A t-interval is always symmetric around the sample mean (xˉ±t∗⋅SE\bar{x} \pm t^* \cdot SE). However, when the underlying data is strongly skewed, the sampling distribution of the mean will also be skewed, especially with a modest sample size like n=35n=35. The bootstrap percentile interval will capture this asymmetry. For left-skewed data, the left tail of the sampling distribution is longer, so the distance from the sample mean to the lower bound of the confidence interval will be greater than the distance to the upper bound. This means the sample mean will be closer to the upper bound.

Question 19

A researcher has an original sample of 30 measurements of stream acidity. To estimate the sampling distribution of the sample median, they decide to use bootstrap resampling. Which of the following correctly describes the process of generating a single bootstrap sample?

  1. Randomly select 30 measurements from the original sample without replacement.
  2. Randomly select 30 measurements from the original sample with replacement. (correct answer)
  3. Fit a theoretical distribution, such as a normal distribution, to the original sample and then draw 30 measurements from it.
  4. Calculate the median of the original sample and add random noise drawn from a standard normal distribution to create 30 new values.
Explanation: The core principle of bootstrap resampling is to simulate the process of drawing samples from a population by instead drawing samples from the original sample itself. This process must be done with replacement to allow for variability in the bootstrap samples. The bootstrap sample size should be the same as the original sample size. Choice B correctly describes this procedure. Choice A describes a permutation, not a bootstrap sample. Choices C and D describe other simulation methods (parametric bootstrap and general Monte Carlo methods, respectively), not the standard non-parametric bootstrap.

Question 20

A bootstrap distribution of 5,000 sample means is generated from an original sample of size n=40n=40. The mean of the original sample was 112.5. The resulting bootstrap distribution of means appears approximately normal. The center of this bootstrap distribution will be closest to which value?

  1. The mean of the population from which the original sample was drawn.
  2. The mean of the original sample, 112.5. (correct answer)
  3. 0, as the distribution is centered for standardization.
  4. The median of the original sample.
Explanation: A key property of the bootstrap distribution is that it is centered at the statistic from the original sample. Since resampling is done from the original sample, the average of the bootstrap sample statistics will converge to the original sample's statistic. It is an approximation of the sampling distribution centered at the sample statistic, not the (unknown) population parameter. Therefore, the center will be very close to the original sample mean of 112.5.