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Chapter 8: Problem 1

Explain the difference between a population characteristic and a statistic.

Short Answer

Expert verified
A population characteristic (or parameter) is a descriptor of an entire population, while a statistic only describes a subset or sample of the population. The values of population characteristics are fixed, while statistics can vary depending on the sample taken from the population.

Step by step solution

01

Define a population characteristic

A population characteristic, also known as a parameter, is a value that accurately represents a particular aspect of an entire population. For example, the average salary of every employee in a country is a population characteristic.
02

Define a statistic

A statistic is a value that represents a particular aspect of a sample, which is a subset of the population. For example, if you take the average salary of a group of 100 employees from the country, that would be a statistic.
03

Discuss differences

The main difference between a population characteristic and a statistic is that a characteristic is a descriptor of an entire population, while a statistic only describes a portion or sample of the population. Population characteristics are fixed and do not change unless the population changes, while statistics vary depending on the sample.

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Most popular questions from this chapter

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 proportion of all subscribers who watch it is only \(.20\) ?

Consider the following population: \(\\{2,3,3,4,4\\}\). The value of \(\mu\) is \(3.2\), but suppose that this is not known to an investigator, who therefore wants to estimate \(\mu\) from sample data. Three possible statistics for estimating \(\mu\) are Statistic \(1:\) the sample mean, \(\bar{x}\) Statistic \(2:\) the sample median Statistic \(3 :\) the average of the largest and the smallest values in the sample A random sample of size 3 will be selected without replacement. Provided that we disregard the order in which the observations are selected, there are 10 possible samples that might result (writing 3 and \(3^{*}, 4\) and \(4^{*}\) to distinguish the two 3's and the two t's in the population): $$ \begin{array}{lllll} 2,3,3^{*} & 2,3,4 & 2,3,4^{*} & 2,3^{*}, 4 & 2,3^{*}, 4^{*} \\ 2,4,4^{*} & 3,3^{*}, 4 & 3,3^{*}, 4^{*} & 3,4,4^{*} & 3^{*}, 4,4^{*} \end{array} $$ For each of these 10 samples, compute Statistics 1,2 , and 3. Construct the sampling distribution of each of these statistics. Which statistic would you recommend for estimating \(\mu\) and why?

The thickness (in millimeters) of the coating applied to disk drives is one 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 \(2 \mathrm{~mm}\) 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 \(2 \mathrm{~mm}\), the desired value. An \(\bar{x}\) value farther from 2 than \(3 \sigma_{\bar{x}}\) is interpreted as an indication of a problem that needs attention. Compute \(2 \pm 3 \sigma_{\bar{x}} .\) (A plot over time of \(\bar{x}\) values with horizontal lines drawn at the limits \(\mu \pm 3 \sigma_{x}^{-}\) is called a process control chart.) c. Referring to Part (b), what is the probability that a sample mean will be outside \(2 \pm 3 \sigma_{\bar{x}}^{-}\) just by chance (that is, 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 \(2.05 \mathrm{~mm}\). What is the probability that a problem will be detected when the next sample is taken? (Hint: This will occur if \(\bar{x}>2+3 \sigma_{\bar{x}}\) or \(\bar{x}<2-3 \sigma_{\bar{x}}\) when \(\left.\mu=2.05 .\right)\)

The article 鈥淯nmarried Couples More Likely to Be Interracial" (San Luis Obispo Tribune, March 13 ,2002) reported that \(7 \%\) of married couples in the United States are mixed racially or ethnically. Consider the population consisting of all married couples in the United States. a. A random sample of \(n=100\) couples will be selected from this population and \(\hat{p}\), the proportion of couples that are mixed racially or ethnically, will be computed. What are the mean and standard deviation of the sampling distribution of \(\hat{p}\) ? b. Is it reasonable to assume that the sampling distribution of \(\hat{p}\) is approximately normal for random samples of size \(n=100\) ? Explain. c. Suppose that the sample size is \(n=200\) rather than \(n=100\), as in Part (b). Does the change in sample size change the mean and standard deviation of the sampling distribution of \(\hat{p}\) ? If so, what are the new values for the mean and standard deviation? If not, explain why not. d. Is it reasonable to assume that the sampling distribution of \(\hat{p}\) is approximately normal for random samples of size \(n=200 ?\) Explain. e. When \(n=200\), what is the probability that the proportion of couples in the sample who are racially or ethnically mixed will be greater than 10 ?

For each of the following statements, identify the number that appears in boldface type as the value of either a population characteristic or a statistic: a. A department store reports that \(84 \%\) of all customers who use the store's credit plan pay their bills on time. b. A sample of 100 students at a large university had a mean age of \(24.1\) years. c. The Department of Motor Vehicles reports that \(22 \%\) of all vehicles registered in a particular state are imports. d. A hospital reports that based on the 10 most recent cases, the mean length of stay for surgical patients is \(6.4\) days. e. A consumer group, after testing 100 batteries of a certain brand, reported an average life of 63 hours of use.
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