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Chapter 7: Problem 12

How does the value of \(\sigma_{\bar{x}}\) change as the sample size increases? Explain.

Short Answer

Expert verified
As the sample size increases, the value of the standard error of the mean (\(\sigma_{\bar{x}}\)) decreases, meaning that the sample mean is likely to be closer to the population mean.

Step by step solution

01

Understand the standard error of the mean

The standard error of the mean (\(\sigma_{\bar{x}}\)) is a measure of how spread out the means of different samples from the same population are likely to be. The smaller the standard error, the more closely the sample mean is likely to represent the population mean.
02

Refer to the formula

The formula for the standard error of the mean is \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size. This formula tells us how \(\sigma_{\bar{x}}\) is related to the sample size \(n\).
03

Analyze the relationship

As the denominator \(\sqrt{n}\) in the formula increases (that is, as the sample size \(n\) increases), the value of \(\sigma_{\bar{x}}\)鈥攖he standard error of the mean鈥攄ecreases. This is because you are dividing by a larger number. The smaller \(\sigma_{\bar{x}}\) is, the less the sample mean is likely to be spread out from the population mean.

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Most popular questions from this chapter

According to Moebs Services Inc., an individual checking account at U.S. community banks costs these banks between $$\$ 175$$ and $$\$ 200$$ per year (Time, November 21,2011 ). Suppose that the average annual cost of all such checking accounts at U.S. community banks is $$\$ 190$$ with a standard deviation of $$\$ 20.$$ Find the probability that the average annual cost of a random sample of 100 such checking accounts at U.S. community banks is a. less than $$\$ 187$$ b. more than $$\$ 193.5$$ c. $$\$ 191.70$$ to $$194.5$$

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 \(3.11\)

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Let \(\hat{p}\) be the proportion of elements in a sample that possess a characteristic. a. What is the mean of \(\hat{p}\) ? b. What is the formula to calculate the standard deviation of \(\hat{p} ?\) Assume \(n / N \leq .05\). c. What condition(s) must hold true for the sampling distribution of \(\hat{p}\) to be approximately normal?
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