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 7.4.8a

Chapter 7, Problem 7.4.8a

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

Verified step by step guidance

Step 1: Understand the problem. We are tasked with constructing an 80% confidence interval for the population mean using the given ten bootstrap samples. Bootstrap samples are resampled datasets created from the original data, and they allow us to estimate the sampling distribution of a statistic.

Step 2: Calculate the mean of each of the ten bootstrap samples. For each bootstrap sample, sum all the values and divide by the number of values in that sample. This will give you ten bootstrap sample means.

Step 3: Arrange the ten bootstrap sample means in ascending order. This step is necessary to identify the range of values that will form the confidence interval.

Step 4: Determine the lower and upper bounds of the 80% confidence interval. Since the confidence level is 80%, the remaining 20% is split equally between the lower and upper tails of the distribution. This means we exclude the lowest 10% and the highest 10% of the bootstrap sample means. For ten bootstrap means, this corresponds to excluding the smallest and largest values.

Step 5: Report the confidence interval. The 80% confidence interval is the range between the second smallest and second largest bootstrap sample means. This interval provides an estimate of the population mean based on the bootstrap resampling method.

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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 observed data. This method allows for the creation of multiple simulated samples, which can help in estimating properties like the mean or variance of a population when the original sample size is small or when the population 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 provides an estimate of uncertainty around a sample statistic, indicating how much the sample mean might vary from the actual population mean.

Population Mean

The population mean is the average of a set of values in a population, representing a central point of the data. It is a key parameter in statistics, as it summarizes the overall behavior of the population. In the context of confidence intervals, estimating the population mean helps in understanding the general trend of the data being analyzed.

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Population Standard Deviation Known

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Textbook Question

E-Cigarettes A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.7% of them being regular users of e-cigarettes. Because e-cigarette use is relatively new, there is a need to obtain today’s usage rate. How many adults must be surveyed now if we want a confidence level of 95% and a margin of error of 1.5 percentage points?

a. Assume that nothing is known about the rate of e-cigarette usage among adults.

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

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

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.

Textbook Question

Archeology Archeologists have studied sizes of Egyptian skulls in an attempt to determine whether breeding occurred between different cultures. Listed below are the widths (mm) of skulls from 150 A.D. (based on data from Ancient Races of the Thebaid by Thomson and Randall-Maciver).

a. Use 1000 bootstrap samples to construct a 99% confidence interval estimate of the mean skull width.