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Chapter 32: Problem 17

We plan to use the bootstrap method to construct a confidence interval for a population median from a sample of 43 subjects from the population. An important assumption for using the bootstrap method is (a) the sample is a random sample from the population. (b) the sampling distribution for the sample median must not be well approximated by the Normal distribution. (c) there are no outliers in the sample.

Short Answer

Expert verified
The important assumption is that the sample is a random sample from the population (option a).

Step by step solution

01

Understand the Bootstrap Method

The bootstrap method is a resampling technique used to estimate statistics on a population by sampling a dataset with replacement. This method is particularly useful when the underlying distribution of the data is unknown or when the sample size is small.
02

Evaluate Assumptions of Bootstrap

The assumptions for the bootstrap method include: (a) the sample should be a random sample from the population, so it is representative of the population, (b) the bootstrap method does not rely on the sampling distribution being approximated by a Normal distribution, and (c) outliers can affect the results, but they do not necessarily invalidate the use of the bootstrap method.
03

Assess Each Option

Option (a) states that the sample is a random sample from the population, which is a correct assumption for the bootstrap method as it ensures the sample is representative. Option (b) talks about the Normal distribution, which is not a necessary requirement for bootstrap as it does not depend on distribution shape. Option (c) mentions the presence of outliers, which do not invalidate bootstrap but can affect results.
04

Conclusion

Given these options, the key assumption for using the bootstrap method is that the sample must be a random sample from the population. This ensures that the sample can provide a reliable estimate of the population parameter.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval
A confidence interval is a range of values that estimates an unknown population parameter, such as a median or mean. It is expressed with a degree of certainty, known as the confidence level, which is usually set at 95% or 99%. This means we are 95% or 99% confident that the interval contains the true population parameter.
A confidence interval gives us an idea of the precision and reliability of our estimate. In the context of the bootstrap method, the confidence interval is constructed by taking repeated samples (with replacement) and calculating the statistic of interest for each sample. This process generates a distribution of the statistic, from which the interval is derived.
Bootstrap confidence intervals are advantageous because they do not require the assumption of normally distributed data, making them versatile for different types of data with unknown distributions.
Population Median
The population median is the middle value in a population data set, separating the higher half from the lower half. It is a robust statistic because it is not affected by extreme values or outliers, unlike the mean. When using the bootstrap method, the goal might often be to estimate the population median from a sample of data.
By constructing a confidence interval for the population median, you get an understanding of where the true median may lie within a range of values. This is especially useful when the distribution of the data is skewed or contains outliers, as the median provides a more accurate measure of central tendency than the mean.
Random Sample
A random sample is a subset of a population selected such that each member of the population has an equal chance of being chosen. This is critical in statistical analysis because it ensures that the sample represents the population accurately, and the results can be generalized to the whole population.
In the context of the bootstrap method, having a random sample is crucial because it forms the basis for deriving the confidence interval. If the sample is not random, it might lead to biased estimates, thus compromising the reliability of the bootstrap confidence interval. Ensuring randomness includes avoiding patterns in selection and addressing potential biases, which can result in skewed data representation.
Resampling Technique
The resampling technique, specifically the bootstrap method, involves repeatedly sampling with replacement from an original data set to create "bootstrap samples." These samples are used to estimate the sampling distribution of a statistic.
Resampling techniques are powerful because they rely on the data itself rather than specific distributional assumptions. This makes them particularly useful in situations where the underlying distribution is unknown or complex. By creating many resamples, analysts can calculate numerous estimates of a statistical measure (such as the median) and construct a confidence interval around those estimates.
Bootstrap resampling is widely used due to its flexibility and ease of use, allowing statisticians to obtain estimates and confidence intervals even with small sample sizes or non-standard data distributions.

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Most popular questions from this chapter

Does taking notes by hand in a statistics course improve performance? Some recent research suggests that this may be the case. 2 To explore this, six volunteers (Doug, Elizabeth, Oksana, Sebastian, Vishal, and Xinyi) agree to take part in an experiment. Four are assigned completely at random to take handwritten notes in class, and the other two are assigned to take notes on their laptops. Total points earned on the two in-class exams and final exam are used to determine course performance. The results are (out of a possible total of 500 points): $$ \begin{array}{ll} \hline \text { Handwritten Notes (Person) } & \text { Notes on Laptop (Person) } \\ \hline 380 \text { (Doug) } & 370 \text { (Elizabeth) } \\ \hline 400 \text { (Oksana) } & 310 \text { (Xinyi) } \\ \hline 420 \text { (Sebastian) } & \\ \hline 360 \text { (Vishal) } & \\ \hline \end{array} $$ (a) There are 15 possible ways the six subjects can be assigned to the two groups, with the handwritten notes group having size 4 and the laptop notes group size 2. List these. (b) For each, determine the difference in mean points (mean number of points for the handwritten notes group minus mean number of points for the laptop notes group). Combine any duplicates and make a table of the possible mean differences and the corresponding probability of each under the null hypothesis of no difference in the effect of the treatments on total points earned. (Each of the 15 possible assignments of subjects to treatments has probability \(1 / 15\) under the null hypothesis.) This is the permutation distribution. (c) Compute the \(P\)-value of the data. Assume the two-sided alternative hypothesis is that the mean number of points is different for the two groups. (d) In this example, is it possible to demonstrate significance at the \(5 \%\) level using the permutation test? Explain. (e) Assume that total number of points is Normally distributed for both groups. Use the two-sample \(t\) procedure to test the hypotheses. Use Option 1 if you have access to software.

Our bodies have a natural electrical field that is known to help wounds heal. Does changing the field strength slow healing? A series of experiments with newts investigated this question. In one experiment, the two hind limbs of four newts were assigned at random to either experimental or control groups. This is a matched pairs design. The electrical field in the experimental limbs was reduced to zero by applying a voltage. The control limbs were left alone. Here are the rates at which new cells closed a razor cut in each limb, in micrometers per hour: \({ }^{11}\) $$ \begin{array}{l|cccc} \hline \text { Newt } & 1 & 2 & 3 & 4 \\ \hline \text { Control limb } & 36 & 41 & 39 & 42 \\ \hline \text { Experimental limb } & 28 & 31 & 27 & 33 \\ \hline \end{array} $$ The number of possible random assignments of treatments to the different matched pairs is (a) 4 . (b) \(8 .\) (c) \(16 .\)

The changing climate will probably bring more rain to California, but we don't know whether the additional rain will come during the winter wet season or extend into the long dry season in spring and summer. Kenwyn Suttle of the University of California at Berkeley and his coworkers carried out a randomized controlled experiment to study the effects of more rain in either season. They randomly assigned 12 plots of open grassland to two treatments: added water equal to \(20 \%\) of annual rainfall during January to March (winter) or no added water (control). One response variable was total plant biomass, in grams per square meter, produced in a plot over a year. \({ }^{10}\) Here are data for 2004 (mass in grams per square meter): $$ \begin{array}{ll} \hline \text { Winter } & \text { Control } \\ \hline 254.6453 & 178.9988 \\ \hline 233.8155 & 205.5165 \\ \hline 253.4506 & 242.6795 \\ \hline 228.5882 & 231.7639 \\ \hline 158.6675 & 134.9847 \\ \hline 212.3232 & 212.4862 \\ \hline \end{array} $$ We wish to test whether there is a difference in mean biomass between the two treatment groups. Which of the following is true? (a) This is a randomized controlled experiment, hence a permutation test is more appropriate than a \(t\) test. (b) This is a randomized controlled experiment, and we should try both the permutation test and the \(t\) test and always report only the one with the smaller \(P\)-value. (c) We might prefer using a permutation test for these data rather than a \(t\) test, because the sample sizes are small and the data contain some possible outliers.

The composition of the earth's atmosphere may have changed over time. To try to discover the nature of the atmosphere long ago, we can examine the gas in bubbles inside ancient amber. Amber is tree resin that has hardened and been trapped in rocks. The gas in bubbles within amber should be a sample of the atmosphere at the time the amber was formed. Measurements on specimens of amber from the late Cretaceous era (75 million to 95 million years ago) give these percentages of nitrogen: \({ }^{15} _{\text { }}\) $$ \begin{array}{lllllllll} 63.4 & 65.0 & 64.4 & 63.3 & 54.8 & 64.5 & 60.8 & 49.1 & 51.0 \end{array} $$ Assume (this is not yet agreed on by experts) that these observations are an SRS from the late Cretaceous atmosphere. (a) Construct a 95\% bootstrap confidence interval for the mean percentage of nitrogen in ancient air (the population). (b) We wonder if ancient air differs significantly from the present atmosphere, which is \(78.1 \%\) nitrogen. Based on your confidence interval in part (a), what do you conclude?

We select a random sample of six freshman students from the University of California at Santa Cruz and find that their verbal GREs are 480, 510, 590, 670, 520 , and 630 . Which of the following is not a possible bootstrap sample? (a) \(480,480,480,480,480,480\) (b) \(480,480,480,670,670,670\) (c) \(480,630,630,740,590,510\)
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