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Chapter 8: Problem 2

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

Short Answer

Expert verified
\(\bar{x}\) is the sample mean and \(\mu\) is the population mean - avg. of a sample data set Vs the avg. of an entire population. \(s\) is sample standard deviation and \(\sigma\) is the population standard deviation, measure of the spread of sample data points around the sample mean Vs measure of spread of all data points around the population mean.

Step by step solution

01

Understand Sample Mean and Population Mean

\(\bar{x}\) represents the 'sample mean', which is the average of a set of observations drawn from a larger population. The formula to calculate \(\bar{x}\) is \(\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i\), where \(x_i\) are the data points and \(n\) is the number of data points. \(\mu\) represents the 'population mean', which is the average of the entire population. The formula to calculate \(\mu\) is \(\mu = \frac{1}{N}\sum_{i=1}^{N} x_i\), where \(N\) is the total number of the population.
02

Understand Sample Standard Deviation and Population Standard Deviation

\(s\) represents the 'sample standard deviation', which describes the spread of data points around the sample mean in a set of observations from a population. The formula to calculate \(s\) is \(s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n} (x_i - \bar{x})^2}\), where \(\bar{x}\) is the sample mean. \(\sigma\) represents the 'population standard deviation', which describes the spread of data points around the population mean in the entire population. The formula to calculate \(\sigma\) is \(\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N} (x_i - \mu)^2}\), where \(\mu\) is the population mean.

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Most popular questions from this chapter

Explain the difference between a population characteristic and a statistic.

The thickness (in millimeters) of the coating applied to disk drives is a characteristic that determines the usefulness of the product. When no unusual circumstances are present, the thickness \((x)\) has a normal distribution with a mean of \(3 \mathrm{~mm}\) and a standard deviation of \(0.05\) \(\mathrm{mm}\). Suppose that the process will be monitored by selecting a random sample of 16 drives from each shift's production and determining \(\bar{x}\), the mean coating thickness for the sample. a. Describe the sampling distribution of \(\bar{x}\) (for a sample of size 16 ). b. When no unusual circumstances are present, we expect \(\bar{x}\) to be within \(3 \sigma_{\bar{x}}\) of \(3 \mathrm{~mm}\), the desired value. An \(\bar{x}\) value farther from 3 than \(3 \sigma_{\bar{x}}\) is interpreted as an indication of a problem that needs attention. Compute \(3 \pm 3 \sigma_{\bar{x}}\). (A plot over time of \(\bar{x}\) values with horizontal lines drawn at the limits \(\mu \pm 3 \sigma_{\bar{x}}\) is called a process control chart.) c. Referring to Part (b), what is the probability that a sample mean will be outside \(3 \pm 3 \sigma_{\bar{x}}\) just by chance (i.e., when there are no unusual circumstances)? d. Suppose that a machine used to apply the coating is out of adjustment, resulting in a mean coating thickness of \(3.05 \mathrm{~mm}\). What is the probability that a problem will be detected when the next sample is taken? (Hint: This will occur if \(\bar{x}>3+3 \sigma_{\bar{x}}\) or \(\bar{x}<3-3 \sigma_{\bar{x}}\) when \(\mu=\) 3.05.) b. When no unusual circumstances are present, we expect \(\bar{x}\) to be within \(3 \sigma_{\bar{x}}\) of \(3 \mathrm{~mm}\), the desired value. An \(\bar{x}\) value farther from 3 than \(3 \sigma_{\bar{x}}\) is interpreted as an indication of a problem that needs attention. Compute \(3 \pm 3 \sigma_{\bar{x}}\). (A plot over time of \(\bar{x}\) values with horizontal lines drawn at the limits \(\mu \pm 3 \sigma_{\bar{x}}\) is called a process control chart.)

A manufacturer of computer printers purchases plastic ink cartridges from a vendor. When a large shipment is received, a random sample of 200 cartridges is selected, and each cartridge is inspected. If the sample proportion of defective cartridges is more than \(.02\), the entire shipment is returned to the vendor. a. What is the approximate probability that a shipment will be returned if the true proportion of defective cartridges in the shipment is .05? b. What is the approximate probability that a shipment will not be returned if the true proportion of defective cartridges in the shipment is .10?

Newsweek (November 23, 1992) reported that 40\% of all U.S. employees participate in "self-insurance" health plans \((\pi=.40)\). a. In a random sample of 100 employees, what is the approximate probability that at least half of those in the sample participate in such a plan? b. Suppose you were told that at least 60 of the \(100 \mathrm{em}\) ployees in a sample from your state participated in such a plan. Would you think \(\pi=.40\) for your state? Explain.

The time that a randomly selected individual waits for an elevator in an office building has a uniform distribution over the interval from 0 to \(1 \mathrm{~min}\). It can be shown that for this distribution \(\mu=0.5\) and \(\sigma=0.289\). a. Let \(\bar{x}\) be the sample average waiting time for a random sample of 16 individuals. What are the mean and standard deviation of the sampling distribution of \(\bar{x}\) ? b. Answer Part (a) for a random sample of 50 individuals. In this case, sketch a picture of a good approximation to the actual \(\bar{x}\) distribution.
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