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Chapter 7: Problem 60

If all possible samples of the same (large) size are selected from a population, what percentage of all sample proportions will be within \(2.0\) standard deviations \(\left(\sigma_{\hat{p}}\right)\) of the population proportion?

Short Answer

Expert verified
Approximately 95% of all sample proportions will be within \(2.0\) standard deviations \(\left(\sigma_{\hat{p}}\right)\) of the population proportion.

Step by step solution

01

Understanding the Empirical Rule

First, it is important to realize that the Empirical Rule is being applied. According to the rule, about 68% of values draw from a normal distribution are within 1 standard deviation \(\left(\sigma_{\hat{p}}\right)\) away from the mean; about 95% are within 2 standard deviations; and about 99.7% lie within 3 standard deviations. This indicates that if all possible samples of the same size are selected from a population, the distribution of the sample proportions approximates a normal distribution due to the central limit theorem, especially as the exercise specifies a 'large' sample size.
02

Application of the Empirical Rule

Since the question specifically asks for the percentage of all sample proportions that will be within \(2.0\) standard deviations of the population proportion, refer to the above description of the Empirical Rule that 95% of values drew from a normal distribution are within 2 standard deviations away from the mean. Hence, approximately 95% of all sample proportions will be within \(2.0\) standard deviations \(\left(\sigma_{\hat{p}}\right)\) of the population proportion.

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Most popular questions from this chapter

The following data give the ages (in years) of all six members of a family. \(\begin{array}{llllll}55 & 53 & 28 & 25 & 21 & 15\end{array}\) a. Let \(x\) denote the age of a member of this family. Write the population probability distribution of \(x\). b. List all the possible samples of size four (without replacement) that can be selected from this population. Calculate the mean for each of these samples. Write the sampling distribution of \(\bar{x}\). c. Calculate the mean for the population data. Select one random sample of size four and calculate the sample mean \(\bar{x}\). Compute the sampling error.

The amounts of electricity bills for all households in a city have a skewed probability distribution with a mean of \(\$ 140\) and a standard deviation of \(\$ 30\). Find the probability that the mean amount of electric bills for a random sample of 75 households selected from this city will be a. between \(\$ 132\) and \(\$ 136\) b. within \(\$ 6\) of the population mean c. more than the population mean by at least \(\$ 4\)

The amounts of electricity bills for all households in a particular city have an approximate normal distribution with a mean of \(\$ 140\) and a standard deviation of \(\$ 30\). Let \(\bar{x}\) be the mean amount of electricity bills for a random sample of 25 households selected from this city. Find the mean and standard deviation of \(\bar{x}\), and comment on the shape of its sampling distribution.

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

What is an estimator? When is an estimator unbiased? Is the sample mean, \(\bar{x}\), an unbiased estimator of \(\mu\) ? Explain.
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