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Ch. 7 - Estimating Parameters and Determining Sample Sizes
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 7 - Estimating Parameters and Determining Sample SizesProblem 7.4.6
Chapter 7, Problem 7.4.6

Seating Choice In a 3M Privacy Filters poll, respondents were asked to identify their favorite seat when they fly, and the results include these responses: window, window, other, other. Letting “window” and letting “other”, those four responses can be represented as {1, 1, 0, 0}. Here are ten bootstrap samples for those responses: [Image]
Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the proportion of respondents who indicated their favorite seat is “window.”

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Step 1: Understand the problem. The goal is to construct an 80% confidence interval for the proportion of respondents who prefer the 'window' seat using the given bootstrap samples. Each bootstrap sample represents a resampling of the original data with replacement.
Step 2: Calculate the proportion of 'window' responses (represented as 1s) for each bootstrap sample. For each sample, count the number of 1s and divide by the total number of responses in that sample. Use the formula: nwindowN, where nwindow is the count of 'window' responses and N is the total number of responses in the sample.
Step 3: Organize the proportions calculated in Step 2 into a list. These proportions represent the bootstrap estimates of the proportion of 'window' preferences.
Step 4: Determine the 80% confidence interval. To do this, sort the list of bootstrap proportions in ascending order. Identify the lower and upper bounds of the interval by finding the 10th and 90th percentiles of the sorted proportions. This corresponds to the middle 80% of the bootstrap distribution.
Step 5: Report the confidence interval. The interval will be expressed as [lower bound, upper bound], where the lower bound is the 10th percentile and the upper bound is the 90th percentile of the bootstrap proportions. This interval provides an estimate of the proportion of respondents who prefer the 'window' seat with 80% confidence.

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Key Concepts

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

Bootstrap Sampling

Bootstrap sampling is a resampling technique used to estimate the distribution of a statistic by repeatedly sampling with replacement from the original data. This method allows for the creation of multiple simulated samples, which can help in estimating confidence intervals and assessing the variability of a statistic, such as the proportion of respondents favoring a particular choice.
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Sampling Distribution of Sample Proportion

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, such as 80%. It provides an estimate of uncertainty around a sample statistic, allowing researchers to make inferences about the population based on sample data.
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Introduction to Confidence Intervals

Proportion

Proportion is a statistical measure that represents the fraction of a whole, often expressed as a percentage. In the context of survey responses, it indicates the ratio of respondents who selected a particular option (e.g., 'window') compared to the total number of respondents. Understanding proportions is essential for analyzing categorical data and making comparisons.
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Guided course
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Difference in Proportions: Hypothesis Tests
Related Practice
Textbook Question

Second-Hand Smoke Refer to Data Set 15 “Passive and Active Smoke” and construct a 95% confidence interval estimates of the mean cotinine level in each of three samples: (1) people who smoke; (2) people who don’t smoke but are exposed to tobacco smoke at home or work; (3) people who don’t smoke and are not exposed to smoke. Measuring cotinine in people’s blood is the most reliable way to determine exposure to nicotine. What do the confidence intervals suggest about the effects of smoking and second-hand smoke?

Textbook Question

Heights of Presidents Refer to Data Set 22 “Presidents” in Appendix B. Treat the data as a sample and find the proportion of presidents who were taller than their opponents. Use that result to construct a 95% confidence interval estimate of the population percentage. Based on the result, does it appear that greater height is an advantage for presidential candidates? Why or why not?

Textbook Question

Finding Critical Values


In Exercises 5–8, find the critical value z=a/2 that corresponds to the given confidence level.


99.5%

Textbook Question

Professor Evaluation Scores Listed below are student evaluation scores of professors from Data Set 28 “Course Evaluations” in Appendix B. Construct a 95% confidence interval estimate of for each of the two data sets. Does there appear to be a difference in variation?

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Textbook Question

Finding Critical Values.


In Exercises 5–8, find the critical value z=a/2 that corresponds to the given confidence level.


90%

Textbook Question

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?