Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problem 7.4.15a

Chapter 7, Problem 7.4.15a

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.

Verified step by step guidance

Step 1: Understand the problem. The goal is to construct a 95% confidence interval for the population standard deviation (σ) using bootstrap sampling. Bootstrap sampling involves resampling the given data with replacement to create multiple simulated samples.

Step 2: Generate 1000 bootstrap samples. From the given data (heights of female soccer players), repeatedly draw random samples of the same size as the original dataset, with replacement, to create 1000 bootstrap samples.

Step 3: Calculate the standard deviation for each bootstrap sample. For each of the 1000 bootstrap samples, compute the standard deviation using the formula:

\sqrt{\frac{\sum {(x}_{i} - x) 2}{n}}

, where

x

is the mean of the sample,

n

is the sample size, and

x_{i}

are the individual data points.

Step 4: Create a distribution of the bootstrap standard deviations. After calculating the standard deviation for each of the 1000 bootstrap samples, you will have a distribution of standard deviations.

Step 5: Determine the 95% confidence interval. To find the 95% confidence interval for σ, identify the 2.5th percentile and the 97.5th percentile of the bootstrap standard deviation distribution. These percentiles represent the lower and upper bounds of the confidence interval, respectively.

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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 construction of confidence intervals and the estimation of standard errors without relying on traditional parametric assumptions. In this context, it involves creating 1000 samples from the heights of female soccer players to assess variability.

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 expressed as a percentage. For example, a 95% confidence interval suggests that if the same sampling process were repeated multiple times, approximately 95% of the calculated intervals would contain the true parameter. In this case, it pertains to estimating the standard deviation (σ) of the heights.

Standard Deviation (σ)

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much individual data points differ from the mean of the dataset. In the context of the question, estimating σ using bootstrap samples helps to understand the variability in the heights of female soccer players, which is crucial for making inferences about the broader population of professional women soccer players.

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Calculating Standard Deviation

Related Practice

Textbook Question

Smart Phone Apple is planning for the launch of a new and improved iPhone. The marketing team wants to know the worldwide percentage of consumers who intend to purchase the new model, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage?

a. Assume that nothing is known about the worldwide percentage of consumers who intend to buy the new model.

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Textbook Question

Large Data Sets from Appendix B. In Exercises 21 and 22, use the data set in Appendix B. Assume that each sample is a simple random sample obtained from a population with a normal distribution.

Birth Weights Refer to Data Set 6 “Births” in Appendix B.

a. Use the 205 birth weights of girls to construct a 95% confidence interval estimate of the standard deviation of the population from which the sample was obtained.

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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:

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

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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?

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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?

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Textbook Question

b. In a study of failure rates of computer hard drives, 45 Toshiba model MD04ABA500V hard drives were tested and there were no failures. What is the 95% upper bound for the percentage of failures for the population of all such hard drives?

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