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?

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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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