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Chapter 7: Problem 29

If all possible samples of the same (large) size are selected from a population, what percentage of all the sample means will be within \(1.5\) standard deviations \(\left(\sigma_{\bar{x}}\right)\) of the population mean?

Short Answer

Expert verified
Approximately 81.5% of all sample means will be within 1.5 standard deviations of the population mean.

Step by step solution

01

Understand the Empirical Rule

The empirical rule, also known as the 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean: 68% falls within the first standard deviation, 95% within two, and 99.7% within three.
02

Relate Empirical Rule to the Problem

The question asks for the percentage of sample means that would fall within 1.5 standard deviations of the population mean. We can't directly read this off from the Empirical Rule, but we can interpolate between the percentages given.
03

Interpolation of Percentage

By the empirical rule, 68% of sample means will fall within 1 standard deviation (\(\sigma_{\bar{x}}\)) of the population mean, while 95% of all sample means are expected to fall within 2 standard deviations. We can estimate the percentage within 1.5 standard deviations by taking a halfway point between 68% and 95%. This gives us approximately 81.5%.

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Most popular questions from this chapter

The GPAs of all students enrolled at a large university have an approximately normal distribution with a mean of \(3.02\) and a standard deviation of \(.29\). Find the probability that the mean GPA of a random sample of 20 students selected from this university is a. \(3.10\) or higher b. \(2.90\) or lower c. \(2.95\) to \(3.11\)

According to the central limit theorem, the sampling distribution of \(\hat{p}\) is approximately normal when the sample is large. What is considered a large sample in the case of the proportion? Briefly explain.

Briefly explain the meaning of a population probability distribution and a sampling distribution. Give an example of each.

What is the estimator of the population proportion? Is this estimator an unbiased estimator of \(p ?\) Explain why or why not.

According to a Gallup poll conducted April \(3-6,2014,21 \%\) of Americans aged 18 to 29 said that college loans and/or expenses were the top financial problem facing their families. Assume that this percentage is true for the current population of Americans aged 18 to \(29 .\) Let \(\hat{p}\) be the proportion in a random sample of 900 Americans aged 18 to 29 who hold the above opinion. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\) and describe its shape.
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