Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problem 7.4.20

Chapter 7, Problem 7.4.20

Medical Malpractice In a study of 1228 randomly selected medical malpractice lawsuits, it was found that 856 of them were dropped or dismissed (based on data from the Physicians Insurers Association of America). Use the bootstrap method to construct a 95% confidence interval estimate of the proportion of lawsuits that are dropped or dismissed. Use 1000 bootstrap samples. How does the result compare to the confidence interval found in Exercise 16 “Medical Malpractice” from Section 7-1?

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Step 1: Understand the problem. We are tasked with constructing a 95% confidence interval for the proportion of medical malpractice lawsuits that are dropped or dismissed using the bootstrap method. The given data includes 1228 lawsuits, of which 856 were dropped or dismissed.

Step 2: Calculate the observed proportion (p̂) of lawsuits that were dropped or dismissed. This is done using the formula:

\frac{856}{1228}

. This proportion will serve as the basis for generating bootstrap samples.

Step 3: Generate 1000 bootstrap samples. For each bootstrap sample, randomly resample the original dataset (with replacement) to create a new dataset of the same size (1228 lawsuits). For each sample, calculate the proportion of lawsuits that are dropped or dismissed.

Step 4: Compute the 95% confidence interval from the bootstrap samples. Sort the 1000 bootstrap proportions in ascending order. Identify the 2.5th percentile and the 97.5th percentile of the sorted proportions. These values represent the lower and upper bounds of the confidence interval, respectively.

Step 5: Compare the bootstrap confidence interval to the one found in Exercise 16 from Section 7-1. Analyze whether the intervals overlap, differ in width, or provide similar estimates of the proportion. Discuss the implications of using the bootstrap method versus traditional methods for constructing confidence intervals.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bootstrap Method

The bootstrap method is a resampling technique used to estimate the distribution of a statistic by repeatedly sampling with replacement from the observed data. This approach allows for the construction of confidence intervals and hypothesis testing without relying on strong parametric assumptions. In this case, it will help estimate the proportion of dropped or dismissed lawsuits by generating multiple samples from the original dataset.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95%. It provides an estimate of uncertainty around the sample proportion, indicating how much the sample result might vary if different samples were taken. Understanding how to interpret and calculate confidence intervals is crucial for assessing the reliability of statistical estimates.

Proportion

Proportion is a statistical measure that represents the fraction of a whole, often expressed as a percentage. In this context, it refers to the ratio of medical malpractice lawsuits that were dropped or dismissed to the total number of lawsuits studied. Calculating the proportion is essential for understanding the prevalence of this outcome in the sample and for constructing the confidence interval around this estimate.

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