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Use technology to create the large number of bootstrap samples. Weights of respondents were recorded as part of the California Health Interview Survey. The last digits of weights from 50 randomly selected respondents are below. $$\begin{array}{ccccccccccccccccccccc}5 & 0 & 1 & 0 & 2 & 0 & 5 & 0 & 5 & 0 & 3 & 8 & 5 & 0 & 5 & 0 & 5 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 8 \\\5 & 5 & 0 & 4 & 5 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 9 & 5 & 3 & 0 & 5 & 0 & 0 & 0 & 5 & 8\end{array}$$ a. Use the bootstrap method with 1000 bootstrap samples to find a \(95 \%\) confidence interval estimate of \(\sigma\). b. Find the \(95 \%\) confidence interval estimate of \(\sigma\) found by using the methods of Section \(7-3 .\) c. Compare the results. If the two confidence intervals are different, which one is better? Why?

Step by step solution

01

- Understand the Data

The given data consists of last digits of weights from 50 respondents. These digits range from 0 to 9 and represent a sample we can use to apply the bootstrap method and statistical calculations.

02

- Generate Bootstrap Samples

To apply the bootstrap method, generate 1000 bootstrap samples from the given data. Each bootstrap sample should be created by randomly sampling with replacement from the original data set of 50 digits.

03

- Calculate Standard Deviation for Each Bootstrap Sample

For each of the 1000 bootstrap samples, calculate the standard deviation. This will give us a distribution of standard deviations based on the bootstrap samples.

04

- Determine the Bootstrap Confidence Interval

From the distribution of the 1000 standard deviations, determine the 2.5th and 97.5th percentiles. These percentiles provide the bounds for the 95% confidence interval for the standard deviation (\( \sigma \)).

05

- Apply Section 7-3 Method

Using the methods outlined in Section 7-3, calculate the 95% confidence interval directly from the sample standard deviation and sample size. The calculation involves finding the critical values from the chi-square distribution.

06

- Compare the Results

Compare the confidence intervals obtained from the bootstrap method and Section 7-3 method. Discuss which interval might be more reliable based on the robustness of the methods involved and any assumptions they make.

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Key Concepts

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

bootstrap samples

Bootstrap sampling is a powerful statistical method used to create multiple samples from a given dataset. This technique helps in estimating the properties of the population. In our exercise, the data consists of the last digits of weights from 50 respondents. To carry out bootstrap sampling, we need to generate 1000 new samples.

standard deviation

Standard deviation is a measure that indicates the amount of variation or dispersion in a set of values. In our task, once we have the 1000 bootstrap samples, we calculate the standard deviation for each sample. This provides a distribution of standard deviations to work with.

confidence interval

A confidence interval offers a range within which we expect a parameter to lie, based on the data. In this case, we seek a 95% confidence interval for the standard deviation of the dataset. By determining the 2.5th and 97.5th percentiles from our standard deviation distribution, we establish this interval.

statistical methods

The exercise involves comparing two statistical methods. The bootstrap method and the traditional method from Section 7-3, which uses the chi-square distribution. The robustness and assumptions underlying each method can affect the trustworthiness of the confidence intervals they produce. Comparing these, we can determine which method might provide a more reliable estimate.