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Chapter 8: Problem 12

Explain the difference between \(\sigma\) and \(\sigma_{\bar{x}}\) and between \(\mu\) and \(\mu_{\bar{x}}\)

Short Answer

Expert verified
蟽 is the standard deviation of a population and 渭 is the mean of the population. 蟽_{\bar{x}} is the standard deviation of the sample means, also known as the standard error, and 渭_{\bar{x}} is the average of the sample means from the population. Hence, '蟽' and '渭' pertains to the entire population while '蟽_{\bar{x}}' and '渭_{\bar{x}}' pertain to statistical samples taken from the population.

Step by step solution

01

Understanding 蟽 and 蟽_{\bar{x}}

The symbol 蟽 stands for the population standard deviation, it quantifies the amount of variation in a set of data values. While 蟽_{\bar{x}}, also known as the standard error, represents the standard deviation of the distribution of the sample means. It quantifies the precision with which the mean of a sample estimates the mean of the population.
02

Understanding 渭 and 渭_{\bar{x}}

The symbol 渭 is used to represent the population mean, which is the average of all values in the population. On the other hand, 渭_{\bar{x}} is the mean of the sample means, that signifies the average of all the sample means from the population.

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Most popular questions from this chapter

What is the difference between \(\bar{x}\) and \(\mu\) ? between \(s\) and \(\sigma\) ?

Suppose that \(20 \%\) of the subscribers of a cable television company watch the shopping channel at least once a week. The cable company is trying to decide whether to replace this channel with a new local station. A survey of 100 subscribers will be undertaken. The cable company has decided to keep the shopping channel if the sample proportion is greater than \(.25 .\) What is the approximate probability that the cable company will keep the shopping channel, even though the true proportion who watch it is only \(.20 ?\)

Suppose that a sample of size 100 is to be drawn from a population with standard deviation \(10 .\) a. What is the probability that the sample mean will be within 2 of the value of \(\mu\) ? b. For this example \((n=100, \sigma=10)\), complete each of the following statements by computing the appropriate value: i. Approximately 95% of the time, \(\bar{x}\) will be within _____ of \(\mu .\) ii. Approximately 0.3% of the time, \(\bar{x}\) will be farther than _____ from\(\mu .\)

Suppose that a particular candidate for public office is in fact favored by \(48 \%\) of all registered voters in the district. A polling organization will take a random sample of 500 voters and will use \(p\), the sample proportion, to estimate \(\pi\). What is the approximate probability that \(p\) will be greater than .5, causing the polling organization to incorrectly predict the result of the upcoming election?

A random sample is to be selected from a population that has a proportion of successes \(\pi=.65 .\) Determine the mean and standard deviation of the sampling distribution of \(p\) for each of the following sample sizes: a. \(n=10\) b. \(n=20\) c. \(n=30\) d. \(n=50\) e. \(n=100\) f. \(n=200\)
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