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 7.4.7a

Chapter 7, Problem 7.4.7a

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.

Verified step by step guidance

Step 1: Understand the problem. The goal is to construct an 80% confidence interval for the mean weight change using the given bootstrap samples. Bootstrap samples are resampled datasets (with replacement) from the original data, and they allow us to estimate the sampling distribution of a statistic (in this case, the mean).

Step 2: Calculate the mean of each bootstrap sample. For each of the ten bootstrap samples provided, compute the mean weight change. Use the formula for the mean:

μ = \sum x

_in, where

x

_i represents the individual data points in the sample, and

n

is the number of data points in the sample.

Step 3: Arrange the means in ascending order. Once you have calculated the means for all ten bootstrap samples, sort them from smallest to largest. This will help in identifying the percentiles needed for the confidence interval.

Step 4: Determine the percentiles for the 80% confidence interval. For an 80% confidence interval, you need to exclude 10% of the data from each tail of the distribution. This corresponds to the 10th percentile (lower bound) and the 90th percentile (upper bound) of the sorted bootstrap means.

Step 5: Identify the confidence interval bounds. Using the sorted bootstrap means, find the values at the 10th and 90th percentiles. These two values represent the lower and upper bounds of the 80% confidence interval for the mean weight change. Report the interval as the final result.

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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 set. This method allows for the creation of multiple simulated samples, which can help in estimating the variability of a statistic, such as the mean. It is particularly useful when the sample size is small or when the underlying distribution is unknown.

Confidence Interval

A confidence interval is a range of values, derived from a data set, that is likely to contain the true population parameter with a specified level of confidence, such as 80%. It is calculated using the sample mean and the standard error, which accounts for the variability in the sample. The wider the interval, the more uncertainty there is about the estimate, while a narrower interval indicates more precision.

Mean Weight Change

The mean weight change refers to the average change in weight of a group of individuals, calculated by summing all individual weight changes and dividing by the number of individuals. In the context of the 'Freshman 15' study, it provides insight into the overall trend of weight gain or loss among college students during their freshman year. Understanding the mean is crucial for interpreting the data and making inferences about the population.

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Calculating the Mean

Related Practice

Textbook Question

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.

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

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

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

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?

Textbook Question

15. HEIGHTS OF FEMALE SOCCER PLAYERS Listed below are the heights (in.) of players on the U.S. Women’s National Soccer Team (at the time of this writing). Use those heights as a sample of the heights of all professional women soccer players.

a. Use 1000 bootstrap samples to construct a 95% confidence interval estimate of σ.