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 7.4.4

Chapter 7, Problem 7.4.4

Mean Assume that we want to use the sample data given in Exercise 1 with the bootstrap method to estimate the population mean. The mean of the values in Exercise 1 is 54.3 seconds, and the mean of all of the tobacco times in Data Set 20 “Alcohol and Tobacco in Movies” from Appendix B is 57.4 seconds. If we use 1000 bootstrap samples and find the corresponding 1000 means, do we expect that those 1000 means will target 54.3 seconds or 57.4 seconds? What does that result suggest about the bootstrap method in this case?

Verified step by step guidance

Step 1: Understand the bootstrap method. The bootstrap method involves resampling the original sample data with replacement to create multiple bootstrap samples. These samples are then used to calculate statistics, such as the mean, to estimate the population parameter.

Step 2: Identify the key values in the problem. The mean of the sample data is 54.3 seconds, and the mean of the population data (all tobacco times) is 57.4 seconds. The bootstrap method uses the sample data to generate estimates, not the population data.

Step 3: Recognize the target of bootstrap means. Since the bootstrap method relies on resampling the sample data, the 1000 bootstrap means will target the sample mean of 54.3 seconds, not the population mean of 57.4 seconds.

Step 4: Interpret the implication of the bootstrap method. This result suggests that the bootstrap method is effective for estimating the sample mean but does not directly target the population mean unless the sample is representative of the population.

Step 5: Conclude the reasoning. The bootstrap method assumes that the sample data is a good representation of the population. If the sample is biased or not representative, the bootstrap estimates may not accurately reflect the population parameter.

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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 statisticians to assess the variability and confidence intervals of estimates, such as the mean, without relying on traditional parametric assumptions. By generating multiple samples, the bootstrap method provides insights into the stability and reliability of the original sample's statistics.

Population Mean vs. Sample Mean

The population mean is the average of all possible values in a population, while the sample mean is the average calculated from a subset of that population. In the context of the question, the sample mean of 54.3 seconds is derived from the specific data set, whereas the population mean of 57.4 seconds represents a broader context. Understanding the difference between these two means is crucial for interpreting the results of the bootstrap method and its implications for estimating population parameters.

Sampling Distribution

The sampling distribution refers to the probability distribution of a statistic (like the mean) obtained from a large number of samples drawn from the same population. It illustrates how the sample mean varies from sample to sample and is foundational for inferential statistics. In this case, analyzing the 1000 bootstrap means will help determine whether they cluster around the sample mean (54.3 seconds) or the population mean (57.4 seconds), providing insights into the accuracy of the bootstrap estimates.

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Sampling Distribution of Sample Proportion

Related Practice

Textbook Question

Atkins Weight Loss Program In a test of weight loss programs, 40 adults used the Atkins weight loss program. After 12 months, their mean weight loss was found to be 2.1 lb, with a standard deviation of 4.8 lb. Construct a 90% confidence interval estimate of the standard deviation of the weight loss for all such subjects. Does the confidence interval give us information about the effectiveness of the diet?

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Textbook Question

Ages of Moviegoers Find the sample size needed to estimate the mean age of movie patrons, given that we want 98% confidence that the sample mean is within 1.5 years of the population mean. Assume that sigma=19.6 years, based on a previous report from the Motion Picture Association of America. Could the sample be obtained from one movie at one theater?

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Textbook Question

use the given information to find the number of degrees of freedom, the critical values X2L and X2R, and the confidence interval estimate of σ. It is reasonable to assume that a simple random sample has been selected from a population with a normal distribution:

Heights of Men 99% confidence; n=153, s=7.10 cm.

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Textbook Question

One-Sided Confidence Interval A one-sided claim about a population proportion is a claim that the proportion is less than (or greater than) some specific value. Such a claim can be formally addressed using a one-sided confidence interval for p, which can be expressed as p<p+E or p>p-E, where the margin of error E is modified by replacing za/2 with za. (Instead of dividing between two tails of the standard normal distribution, put all of it in one tail.) The Chapter Problem refers to a Sallie Mae survey of 950 undergraduate students, and 53% of the survey subjects take online courses. Use that data to construct a one-sided 95% confidence interval that would be suitable for helping to determine whether the percentage of all undergraduates who take online courses is greater than 50%.

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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.

Measured Results vs. Reported Results The same study cited in the preceding exercise produced these results after six months for the 198 patients given sustained care: 25.8% were no longer smoking, and these results were biochemically confirmed, but 40.9% of these patients reported that they were no longer smoking. Construct the two 95% confidence intervals. Compare the results. What do you conclude?

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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.

Internet Use A random sample of 5005 adults in the United States includes 751 who do not use the Internet (based on three Pew Research Center polls). Construct a 95% confidence interval estimate of the percentage of U.S. adults who do not use the Internet. Based on the result, does it appear that the percentage of U.S. adults who do not use the Internet is different from 48%, which was the percentage in the year 2000?"

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