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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

A random sample is to be selected from a population that has a proportion of successes \(p=.65 .\) Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following sample sizes:\(8.23\) A random sample is to be selected from a population that has a proportion of successes \(p=.65 .\) Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following sample sizes: a. \(n=10\) b. \(n=20\) c. \(n=30\) d. \(n=50\) e. \(n=100\) f. \(n=200\)

Consider the following population: \(\\{1,2,3,4\\} .\) Note that the population mean is $$ \mu=\frac{1+2+3+4}{4}=2.5 $$ a. Suppose that a random sample of size 2 is to be selected without replacement from this population. There are 12 possible samples (provided that the order in which observations are selected is taken into account): $$ \begin{array}{llllll} 1,2 & 1,3 & 1,4 & 2,1 & 2,3 & 2,4 \\ 3,1 & 3,2 & 3,4 & 4,1 & 4,2 & 4,3 \end{array} $$ Compute the sample mean for each of the 12 possible samples. Use this information to construct the sampling distribution of \(\bar{x}\). (Display the sampling distribution as a density histogram.) b. Suppose that a random sample of size 2 is to be selected, but this time sampling will be done with replacement. Using a method similar to that of Part (a), construct the sampling distribution of \(\bar{x}\). (Hint: There are 16 different possible samples in this case.) c. In what ways are the two sampling distributions of Parts (a) and (b) similar? In what ways are they different?

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?

Suppose that a random sample of size 64 is to be selected from a population with mean 40 and standard deviation 5 . a. What are the mean and standard deviation of the \(\bar{x}\) sampling distribution? Describe the shape of the \(\bar{x}\) sampling distribution. b. What is the approximate probability that \(\bar{x}\) will be within \(0.5\) of the population mean \(\mu\) ? c. What is the approximate probability that \(\bar{x}\) will differ from \(\mu\) by more than \(0.7 ?\)

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 \(.20\) ?
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