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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
The value of the standard deviation of the sample mean or \(\sigma_{\bar{x}}\) decreases as the sample size (\(n\)) increases. This is due to the inverse relationship between \(\sigma_{\bar{x}}\) and the square root of the sample size (\(\sqrt{n}\)), as denoted in the formula \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\).

Step by step solution

01

Understanding standard deviation of the sample mean (\(\sigma_{\bar{x}}\))

The standard deviation of the sample mean refers to deviation of the sample means from the population mean. By formula, it is expressed as \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the standard deviation of the population and \(n\) is the size of the sample.
02

Analysing the relation between \(\sigma_{\bar{x}}\) and sample size

From the formula \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), it is clear that \(\sigma_{\bar{x}}\) is inversely related to the square root of sample size \(n\). This means that as the sample size \(n\) increases, the value of \(\sqrt{n}\) also increases, which in turn decreases the value of \(\sigma_{\bar{x}}\). Therefore, the standard deviation of the sample mean (\(\sigma_{\bar{x}}\)) decreases, as the sample size increases.
03

Conclusion

Increasing the sample size reduces the standard deviation of the sample mean. This accounts for greater accuracy in the sample mean representing the population mean, as a large sample size minimizes the found differences or variances among the means of different samples.

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

In an observational study at Turner Field in Atlanta, Georgia, \(43 \%\) of the men were observed not washing their hands after going to the bathroom (Source: Harris Interactive). Assume that this percentage is true for the current population of U.S. men. Let \(\hat{p}\) be the proportion in a random sample of \(110 \mathrm{U} . \mathrm{S}\). men who do not wash their hands after going to the bathroom. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\), and describe its shape.

According to an article on www.PCMag.com, Facebook users spend an average of 190 minutes per month checking and updating their Facebook pages (Source: http://www.pemag.com/article2/ \(0,2817,2342757,00\).asp). Suppose that the current distribution of times spent per month checking and updating their Facebook pages by all users is normally distributed with a mean of 190 minutes and a standard deviation of \(53.4\) minutes. Find the probability that the mean time spent per month checking and updating their Facebook pages by a random sample of 12 Facebook users will be a. within 10 minutes of the population mean b. more than 240 minutes c. at least 20 minutes different than the population mean d. less than 207 minutes

Is the sample proportion a consistent estimator of the population proportion? Explain why or why not.

A company manufactured six television sets on a given day, and these TV sets were inspected for being good or defective. The results of the inspection follow. Good Good \(\quad \begin{array}{llll}\text { Defective } & \text { Defective } & \text { Good } & \text { Good }\end{array}\) r. What proportion of these TV sets are good? b. How many total samples (without replacement) of size five can be selected from this population? c. List all the possible samples of size five that can be selected from this population and calculatc the sample proportion, \(\hat{p}\), of television sets that are good for each sample. Prepare the sampling distribution of \(\hat{p}\). d. For each sample listed in part c. calculate the sampling error.
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