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Chapter 5: Q42E (page 230)

A company maintains three offices in a certain region, each staffed by two employees. Information concerning yearly salaries (\({\rm{1000}}\)s of dollars) is as follows:

\(\begin{array}{*{20}{c}}{{\rm{ Office }}}&{\rm{1}}&{\rm{1}}&{\rm{2}}&{\rm{2}}&{\rm{3}}&{\rm{3}}\\{{\rm{ Employee }}}&{\rm{1}}&{\rm{2}}&{\rm{3}}&{\rm{4}}&{\rm{5}}&{\rm{6}}\\{{\rm{ Salary }}}&{{\rm{29}}{\rm{.7}}}&{{\rm{33}}{\rm{.6}}}&{{\rm{30}}{\rm{.2}}}&{{\rm{33}}{\rm{.6}}}&{{\rm{25}}{\rm{.8}}}&{{\rm{29}}{\rm{.7}}}\end{array}\)

a. Suppose two of these employees are randomly selected from among the six (without replacement). Determine the sampling distribution of the sample mean salary\({\rm{\bar X}}\).

b. Suppose one of the three offices is randomly selected. Let\({{\rm{X}}_{\rm{1}}}\)and\({{\rm{X}}_{\rm{2}}}\)denote the salaries of the two employees. Determine the sampling distribution of\({\rm{\bar X}}\).

c. How does \({\rm{E(\bar X)}}\) from parts (a) and (b) compare to the population mean salary\({\rm{\mu }}\)?

Short Answer

Expert verified

a) The sampling distribution

b) The sampling distribution

c) The population mean salary \({\rm{\mu }}\) is equal to the anticipated values \({\rm{E(\bar X)}}\) from parts (a) and (b).

Step by step solution

01

Definition

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

02

Determine the sampling distribution of the sample mean salary \({\rm{\bar X}}\)

Determine the sample mean salary (sum of all values divided by the number of values) for each sample of \({\rm{2}}\) employees:

Determine the number of times each sample mean appears and the probability associated with it (frequency divided by the total frequency of \({\rm{30}}\)).

03

Determining the sampling distribution of \({\rm{\bar X}}\)

b. Calculate the sample mean pay for each sample comprising one office (sum of all values divided by the number of values):


Determine the number of times each sample mean appears and the probability associated with it (frequency divided by the total frequency of \({\rm{3}}\)).

04

Determine the sampling distribution of the sample mean salary \({\rm{\bar X}}\)

c. The anticipated value is calculated by multiplying each alternative by its probability:

\(\begin{aligned}{\rm{E}}\left( {{{{\rm{\bar X}}}_{\rm{a}}}} \right)\rm &= \sum {\rm{x}} {\rm{P(x)}}\\\rm &= 27{\rm{.75}}\frac{{\rm{4}}}{{{\rm{30}}}}{\rm{ + 28}}\frac{{\rm{2}}}{{{\rm{30}}}}{\rm{ + 29}}{\rm{.7}}\frac{{\rm{6}}}{{{\rm{30}}}}{\rm{ + 29}}{\rm{.95}}\frac{{\rm{4}}}{{{\rm{30}}}}{\rm{ + 31}}{\rm{.65}}\frac{{\rm{8}}}{{{\rm{30}}}}{\rm{ + 31}}{\rm{.9}}\frac{{\rm{4}}}{{{\rm{30}}}}{\rm{ + 33}}{\rm{.6}}\frac{{\rm{2}}}{{{\rm{30}}}}\\&{\rm{\gg 30}}{\rm{.4333}}\\{\rm{E}}\left( {{{{\rm{\bar X}}}_{\rm{b}}}} \right)&{\rm{ = }}\sum {\rm{x}} {\rm{P(x)}}\\&{\rm{ = 27}}{\rm{.75}}\frac{{\rm{1}}}{{\rm{3}}}{\rm{ + 31}}{\rm{.65}}\frac{{\rm{1}}}{{\rm{3}}}{\rm{ + 31}}{\rm{.9}}\frac{{\rm{1}}}{{\rm{3}}}\\&{\rm{\gg 30}}{\rm{.4333}}\end{aligned}\)

The population mean is the sum of all values divided by the number of values:

\({\rm{\mu = }}\frac{{{\rm{29}}{\rm{.7 + 33}}{\rm{.6 + 30}}{\rm{.2 + 33}}{\rm{.6 + 25}}{\rm{.8 + 29}}{\rm{.7}}}}{{\rm{6}}}{\rm{\gg 30}}{\rm{.4333}}\)

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Most popular questions from this chapter

Two components of a minicomputer have the following joint pdf for their useful lifetimes \({\rm{X}}\)and \({\rm{Y}}\)

a. What is the probability that the lifetime \({\rm{X}}\) of the first component exceeds \({\rm{3}}\)?

b. What are the marginal pdf鈥檚 of \({\rm{X}}\)and \({\rm{Y}}\)? Are the two lifetimes independent? Explain.

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A certain market has both an express checkout line and a superexpress checkout line. Let \({{\rm{X}}_{\rm{1}}}\) denote the number of customers in line at the express checkout at a particular time of day, and let \({{\rm{X}}_{\rm{2}}}\) denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of \({{\rm{X}}_{\rm{1}}}\)and\({{\rm{X}}_{\rm{2}}}\) is as given in the accompanying table

a. What is \({\rm{P(}}{{\rm{X}}_{\rm{1}}}{\rm{ = 1}}{{\rm{X}}_{\rm{2}}} = 1)\), that is, the probability that there is exactly one customer in each line?

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d. What is the probability that the total number of customers in the two lines is exactly four? At least four?

A box contains ten sealed envelopes numbered\({\rm{1, \ldots ,10}}\). The first five contain no money, the next three each contains\({\rm{\$ 5}}\), and there is a \({\rm{\$ 10}}\) bill in each of the last two. A sample of size \({\rm{3}}\) is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. If \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and \({{\rm{X}}_{\rm{3}}}\) denote the amounts in the selected envelopes, the statistic of interest is \({\rm{M = }}\) the maximum of\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and\({{\rm{X}}_{\rm{3}}}\).

a. Obtain the probability distribution of this statistic.

b. Describe how you would carry out a simulation experiment to compare the distributions of \({\rm{M}}\) for various sample sizes. How would you guess the distribution would change as \({\rm{n}}\) increases?

Show that if\({\rm{X}}\)and\({\rm{Y}}\)are independent rv's, then\({\rm{E(XY) = E(X) \times E(Y)}}\). Then apply this in Exercise\({\rm{25}}{\rm{.}}\)[A1] (Hint: Consider the continuous case with\({\rm{f(x,y) = }}\)\({{\rm{f}}_{\rm{X}}}{\rm{(x) \times }}{{\rm{f}}_{\rm{Y}}}{\rm{(y)}}\).)

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