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Chapter 7: Problem 28

If all possible samples of the same (large) size are selected from a population, what percentage of all the sample means will be within \(2.5\) standard deviations \(\left(\sigma_{\bar{x}}\right)\) of the population mean?

Short Answer

Expert verified
Approximately 99.38% of all sample means will be within 2.5 standard deviations of the population mean.

Step by step solution

01

Understanding the problem

Here, it is asked what percentage of all the sample means will be within 2.5 standard deviations of the population mean. We know that this matches with the concept of standard normal distribution or Z-distribution.
02

Application of empirical rule

The empirical rule or the 68-95-99.7 rule guides us about the percentage of data lying within certain number of standard deviations. Unfortunately, it does not provide a direct percentage for 2.5 standard deviations.
03

Find exact percentage

To find out the exact percentage, use a standard normal distribution table (Z-table) or use a statistical calculator. In a Z-table for value of 2.5, we find that approximately 99.38% of data lie within +2.5 and -2.5 standard deviations of the mean.

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Most popular questions from this chapter

For a population, \(\mu=125\) and \(\sigma=36\). a. For a sample selected from this population, \(\mu_{\bar{x}}=125\) and \(\sigma_{\bar{x}}=3.6\). Find the sample size. Assume \(n / N \leq .05\). b. For a sample selected from this population, \(\mu_{\bar{x}}=125\) and \(\sigma_{\bar{x}}=2.25\). Find the sample size. Assume \(n / N \leq .05\).

A television reporter is covering the election for mayor of a large city and will conduct an exit poll (interviews with voters immediately after they vote) to make an early prediction of the outcome. Assume that the eventual winner of the election will get \(60 \%\) of the votes. a. What is the probability that a prediction based on an exit poll of a random sample of 25 voters will be correct? In other words, what is the probability that 13 or more of the 25 voters in the sample will have voted for the eventual winner? b. How large a sample would the reporter have to take so that the probability of correctly predicting the outcome would be \(.95\) or higher?

In a population of 18,700 subjects, \(30 \%\) possess a certain characteristic. In a sample of 250 subjects selected from this population, \(25 \%\) possess the same characteristic. How many subjects in the population and sample, respectively, possess this characteristic?

The Toyota Prius hybrid car is estimated to get an average of 50 miles per gallon (mpg) of gas. However, the gas mileage varies from car to car due to a variety of conditions, driving styles, and other factors and has been reported to be as high as \(70 \mathrm{mpg}\). Suppose that the distribution of miles per gallon for Toyota Prius hybrid cars has a mean of \(50 \mathrm{mpg}\) and a standard deviation of \(5.9 \mathrm{mpg}\). Find the probability that the average miles per gallon for 38 randomly selected Prius hybrid cars is a. more than \(51.5\) b. between 48 and 51 c. less than 53 d. greater than the population mean by \(2.5\) or more

If all possible samples of the same (large) size are selected from a population, what percentage of all sample proportions will be within \(3.0\) standard deviations \(\left(\sigma_{\hat{p}}\right)\) of the population proportion?
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