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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

Let\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}{\rm{,}}{{\rm{X}}_{\rm{4}}}{\rm{,}}{{\rm{X}}_{\rm{5}}}\), and \({{\rm{X}}_{\rm{6}}}\) denote the numbers of blue, brown, green, orange, red, and yellow M\&M candies, respectively, in a sample of size\({\rm{n}}\). Then these \({{\rm{X}}_{\rm{i}}}\) 's have a multinomial distribution. According to the M\&M Web site, the color proportions are\({{\rm{p}}_{\rm{1}}}{\rm{ = }}{\rm{.24,}}{{\rm{p}}_{\rm{2}}}{\rm{ = }}{\rm{.13}}\), \({{\rm{p}}_{\rm{3}}}{\rm{ = }}{\rm{.16,}}{{\rm{p}}_{\rm{4}}}{\rm{ = }}{\rm{.20,}}{{\rm{p}}_{\rm{5}}}{\rm{ = }}{\rm{.13}}\), and\({{\rm{p}}_{\rm{6}}}{\rm{ = }}{\rm{.14}}\).

a. If\({\rm{n = 12}}\), what is the probability that there are exactly two M\&Ms of each color?

b. For\({\rm{n = 20}}\), what is the probability that there are at most five orange candies? (Hint: Think of an orange candy as a success and any other color as a failure.)

c. In a sample of\({\rm{20M \backslash Ms}}\), what is the probability that the number of candies that are blue, green, or orange is at least \({\rm{10}}\) ?

Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of \({\rm{\bar X}}\) when the population distribution is lognormal with \({\rm{E(ln(X)) = 3}}\) and\({\rm{V(ln(X)) = 1}}\). Consider the four sample sizes\({\rm{n = 10,20,30}}\), and\({\rm{50}}\), and in each case use \({\rm{1000}}\) replications. For which of these sample sizes does the \({\rm{\bar X}}\) sampling distribution appear to be approximately normal?

The joint probability di\({\rm{Y}} \le {\rm{1) = 0}}{\rm{.12}}\)stribution of the number \({\rm{X}}\) of cars and the number \({\rm{Y}}\) of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table.

a. What is the probability that there is exactly one car and exactly one bus during a cycle?

b. What is the probability that there is at most one car and at most one bus during a cycle?

c. What is the probability that there is exactly one car during a cycle? Exactly one bus?

d. Suppose the left-turn lane is to have a capacity of five cars, and that one bus is equivalent to three cars. What is t\({\rm{p(x,y)}} \ge {\rm{0}}\)e probability of an overflow during a cycle?

e. Are \({\rm{X}}\) and \({\rm{Y}}\) independent rv鈥檚? Explain.

An instructor has given a short quiz consisting of two parts. For a randomly selected student, let \({\rm{X = }}\) the number of points earned on the first part and \({\rm{Y = }}\) the number of points earned on the second part. Suppose that the joint pmf of \({\rm{X}}\) and \({\rm{Y}}\) is given in the accompanying table.

a. If the score recorded in the grade book is the total number of points earned on the two parts, what is the expected recorded score\({\rm{E(X + Y)}}\)?

b. If the maximum of the two scores is recorded, what is the expected recorded score?

Let \({\rm{X}}\)denote the courtship time for a randomly selected female鈥搈ale pair of mating scorpion flies (time from the beginning of interaction until mating). Suppose the mean value of\({\rm{X}}\) is \({\rm{120}}\) min and the standard deviation of \({\rm{X}}\) is \({\rm{110}}\) min (suggested by data in the article 鈥淪hould I Stay or Should I Go? Condition- and Status-Dependent Courtship Decisions in the Scorpion Fly Panorpa Cognate鈥� (Animal Behavior, \({\rm{2009: 491 - 497}}\))). transmitted, there is a \({\rm{ 10\% }}\)chance of a transmission error (a \({\rm{0}}\)becoming a \({\rm{1}}\) or a \({\rm{1}}\)becoming a \({\rm{0}}\)). Assume that bit errors occur independently of one another.

a. Is it plausible that \({\rm{X}}\) is normally distributed?

b. For a random sample of \({\rm{50}}\) such pairs, what is the (approximate) probability that the sample mean courtship time is between \({\rm{100}}\) min and \({\rm{125}}\) min?

c. For a random sample of \({\rm{50}}\) such pairs, what is the (approximate) probability that the total courtship time exceeds \({\rm{150}}\) hrs.?

d. Could the probability requested in (b) be calculated from the given information if the sample size were \({\rm{15}}\) rather than \({\rm{50}}\)? Explain.

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