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Chapter 9: Problem 21

Certain statistics are difficult to bootstrap. One such statistic is the median. Consider the following to see why. (a) Simulate obtaining a random sample of 12 IQ scores. Recall IQ scores are approximately normally distributed with mean 100 and standard deviation \(15 .\) (b) Given that IQ scores are normally distributed, what is the median IQ score? (c) Obtain 1000 bootstrap samples from the data in part (a). Find the median of each bootstrap sample. (d) Draw a histogram of the bootstrap medians from part (c). What do you notice about the distribution? Find the \(95 \%\) confidence interval based on the 1000 bootstrap medians using the percentile method. (e) Repeat parts (a) through (d) using a random sample of 13 IQ scores. (f) Conclude that finding confidence intervals for medians is best if done where the sample size is even.

Short Answer

Expert verified
The median IQ score is 100. The 95% confidence interval from the bootstrap medians provides more reliable results for even-numbers of sampled IQ scores.

Step by step solution

01

Simulate a Sample of IQ Scores

Simulate obtaining a random sample of 12 IQ scores where the IQ scores are approximately normally distributed with mean 100 and standard deviation 15. Use a statistical software or programming language like R, Python, or MATLAB to generate this random sample.
02

Find the Median IQ Score

Given that IQ scores are normally distributed, the median IQ score is equivalent to the mean of the distribution, which is 100.
03

Generate Bootstrap Samples

From the initial random sample of 12 IQ scores, obtain 1000 bootstrap samples by randomly sampling with replacement from the original data set. Each bootstrap sample will also have 12 IQ scores.
04

Calculate Medians of Bootstrap Samples

Calculate the median for each of the 1000 bootstrap samples. Store these median values.
05

Draw Histogram and Find 95% Confidence Interval

Draw a histogram of the 1000 bootstrap medians. Observe the distribution shape. Determine the 95% confidence interval using the percentile method by finding the 2.5th and 97.5th percentiles of the bootstrap medians.
06

Simulate a Sample of 13 IQ Scores

Repeat the entire process, this time starting with a new random sample of 13 IQ scores. Follow the same steps of generating bootstrap samples, calculating medians, drawing the histogram, and determining the 95% confidence interval.
07

Analyze Results for Even and Odd Sample Sizes

Compare the results of the confidence intervals obtained from the samples of 12 and 13 IQ scores. Conclude that finding confidence intervals for medians is more reliable when the sample size is even.

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Key Concepts

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

bootstrap sampling
Bootstrap sampling is a resampling technique used to estimate statistics on a dataset by sampling with replacement. Suppose you have a dataset of 12 IQ scores. To perform bootstrap sampling, you'd randomly select IQ scores from your original dataset to create a new sample of the same size, but with some scores possibly repeated. This process helps in approximating the sampling distribution of a statistic, such as the median.
median estimation
When estimating the median, specifically with normally distributed data, the median is commonly used as it represents the middle value. Given IQ scores are normally distributed with a mean of 100, the median also tends to be 100. Using bootstrap samples, you can determine the median for each sample. By repeating this process across many bootstrap samples, you get a range of median values that helps in estimating the true median.
confidence intervals
Confidence intervals provide a range in which you might expect to find the population parameter, such as the median. To find a 95% confidence interval using bootstrap samples, sort your 1000 median values obtained from each bootstrap sample. The 2.5th percentile of these values gives the lower bound, and the 97.5th percentile provides the upper bound. This method, known as the percentile method, is straightforward and leverages the bootstrap distribution directly.
normal distribution in statistics
The normal distribution is a fundamental concept in statistics, representing a bell-shaped curve where most values cluster around the mean, and probabilities symmetrically decrease as you move toward the extremes. For example, IQ scores tend to follow a normal distribution with a mean of 100 and a standard deviation of 15. Understanding this distribution is crucial for bootstrap sampling and median estimation, allowing for accurate interpretation and analysis of statistical data.

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Most popular questions from this chapter

According to Nielsen/ NetRatings, the mean amount of time spent on MySpace.com per user per month in July 2014 was 171.0 minutes. A \(95 \%\) confidence interval for the mean amount of time spent on MySpace.com monthly has a lower bound of 151.4 minutes and an upper bound of 190.6 minutes. What does this mean?

In a survey of 1008 adult Americans, the Gallup organization asked, "When you retire, do you think you will have enough money to live comfortably or not?" Of the 1008 surveyed, 526 stated that they were worried about having enough money to live comfortably in retirement. Construct a \(90 \%\) confidence interval for the proportion of adult Americans who are worried about having enough money to live comfortably in retirement.

Explain why quadrupling the sample size causes the margin of error to be cut in half.

Construct the appropriate confidence interval. A simple random sample of size \(n=40\) is drawn from a population. The sample mean is found to be \(\bar{x}=120.5,\) and the sample standard deviation is found to be \(s=12.9\). Construct a \(99 \%\) confidence interval for the population mean.

The American Society for Microbiology (ASM) and the Soap and Detergent Association (SDA) jointly commissioned two separate studies, both of which were conducted by Harris Interactive. In one of the studies, 1001 adults were interviewed by telephone and asked about their handwashing habits. In the telephone interviews, 921 of the adults said they always wash their hands in public restrooms. In the other study, the hand-washing behavior of 6076 adults was inconspicuously observed within public restrooms in four U.S. cities and 4679 of the 6076 adults were observed washing their hands. (a) In the telephone survey, what is the variable of interest? Is it qualitative or quantitative? (b) What is the sample in the telephone survey? What is the population to which this study applies? (c) Verify that the requirements for constructing a confidence interval for the population proportion of adults who say they always wash their hands in public restrooms are satisfied. (d) Using the results from the telephone interviews, construct a \(95 \%\) confidence interval for the proportion of adults who say they always wash their hands in public restrooms. (e) In the study where hand-washing behavior was observed, what is the variable of interest? Is it qualitative or quantitative? (f) We are told that 6076 adults were inconspicuously observed, but were not told how these adults were selected. We know randomness is a key ingredient in statistical studies that allows us to generalize results from a sample to a population. Suggest some ways randomness might have been used to select the individuals in this study. (g) Verify the requirements for constructing a confidence interval for the population proportion of adults who actually washed their hands while in a public restroom. (h) Using the results from the observational study, construct a \(95 \%\) confidence interval for the proportion of adults who wash their hands in public restrooms. (i) Based on your findings in parts (a) through (h), what might you conclude about the proportion of adults who say they always wash their hands versus the proportion of adults who actually wash their hands in public restrooms? (j) Cite some sources of variability in both studies.
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