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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 7.4.23a
Chapter 7, Problem 7.4.23a

Analysis of Last Digits Weights of respondents were recorded as part of the California Health Interview Survey. The last digits of weights from 50 randomly selected respondents are listed below.


Last digits of weights from 50 respondents: 5, 0, 1, 0, 2, 0, 5, 0, 5, 0, 3, 8, 5, 0, 5, 5, 6, 0, 0, 0, 0, 8, 5, 5, 0, 4, 5, 0, 4, 0, 0, 0, 0, 8, 0, 9, 5, 3, 0, 5, 0, 0, 5, 8.


a. Use the bootstrap method with 1000 bootstrap samples to find a 95% confidence interval estimate of .

Verified step by step guidance
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Step 1: Understand the bootstrap method. The bootstrap method involves resampling the given data with replacement to create multiple simulated samples. These samples are used to estimate the variability of a statistic, such as the mean or median, and construct confidence intervals.
Step 2: Extract the last digits from the image provided. The last digits are: 5, 0, 1, 0, 2, 0, 5, 0, 5, 0, 3, 8, 5, 0, 5, 6, 0, 0, 0, 0, 0, 8, 5, 5, 0, 4, 5, 0, 4, 0, 0, 0, 0, 8, 0, 9, 5, 3, 0, 5, 0, 0, 0, 5, 8.
Step 3: Generate 1000 bootstrap samples. For each bootstrap sample, randomly select 50 values from the original dataset with replacement. This means some values may appear multiple times in a single sample, while others may not appear at all.
Step 4: Calculate the statistic of interest (e.g., mean, median, etc.) for each of the 1000 bootstrap samples. This will give you a distribution of the statistic based on the resampled data.
Step 5: Construct the 95% confidence interval. Sort the bootstrap statistics in ascending order and identify the values at the 2.5th percentile and the 97.5th percentile of the distribution. These values form the lower and upper bounds of the confidence interval.

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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 estimation of confidence intervals and standard errors without relying on strong parametric assumptions. In this case, it involves creating 1000 bootstrap samples from the last digits of weights to assess the variability and derive a 95% confidence interval.
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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 a sample statistic, indicating how much the sample might differ from the actual population. In this context, the confidence interval will help quantify the uncertainty in estimating the mean or proportion of the last digits of weights.
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Sampling Distribution

The sampling distribution is the probability distribution of a statistic obtained from a large number of samples drawn from a specific population. It describes how the statistic varies from sample to sample and is fundamental in inferential statistics. Understanding the sampling distribution is crucial for applying the bootstrap method, as it allows for the estimation of the variability of the sample statistic and the construction of confidence intervals.
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Related Practice
Textbook Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.


Tennis Challenges In a recent U. S. Open tennis tournament, women playing singles matches used challenges on 137 calls made by the line judges. Among those challenges, 33 were found to be successful with the call overturned.


a. Construct a 99% confidence interval for the percentage of successful challenges.

Textbook Question

Comparing Waiting Lines


The values listed below are waiting times (in minutes) of customers at the Jefferson Valley Bank, where customers enter a single waiting line that feeds three teller windows. Construct a 95% confidence interval for the population standard deviation sigma.

Textbook Question

Cell Phone Radiation Here is a sample of measured radiation emissions (cW/kg) for cell phones (based on data from the Environmental Working Group): 38, 55, 86, 145. Here are ten bootstrap samples:

[Image]


a. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the population mean.


Textbook Question

Freshman 15 Here is a sample of amounts of weight change (kg) of college students in their freshman year (from Data Set 13 “Freshman 15” in Appendix B): 11, 3, 0, , where represents a loss of 2 kg and positive values represent weight gained. Here are ten bootstrap samples:

[Image]


a. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the mean weight change for the population.


Textbook Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.


Job Interviews In a Harris poll of 514 human resource professionals, 90% said that the appearance of a job applicant is most important for a good first impression.


a. Among the 514 human resource professionals who were surveyed, how many of them said that the appearance of a job applicant is most important for a good first impression?


Textbook Question

No Failures According to the Rule of Three, when we have a sample size n with x=0 successes, we have 95% confidence that the true population proportion has an upper bound of 3/n. (See “A Look at the Rule of Three,” by Jovanovic and Levy, American Statistician, Vol. 51, No. 2.)


a. If n independent trials result in no successes, why can’t we find confidence interval limits by using the methods described in this section?