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Chapter 5: Q45E (page 230)

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?

Short Answer

Expert verified

The last histogram and normal probability plot are for sample size\({\rm{n = 300}}\), mainly to show that the sample distribution becomes closer to normal as \({\rm{n}}\) rises.

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

Determining the sample size

Simulate the sample distribution of \({\rm{\bar X}}\) using any programmed, such as \({\rm{R}}\), math lab, python, or a statistical computer package, given a lognormal distribution with expectation \({\rm{3}}\) and variance\({\rm{1}}\).

Calculate the mean for each sample first. A histogram and a normal probability plot should be plotted. The distribution tends to be normal as \({\rm{n}}\) rises; nevertheless, as the histograms and normal probability plots below show, the distribution is not roughly normal for given sample sizes. The last histogram and normal probability plot are for sample size\({\rm{n = 300}}\), mainly to show that the sample distribution becomes closer to normal as \({\rm{n}}\) rises.

For\({\rm{n = 10}}\), the following are histogram and normal probability plot:

03

Determining the sample size

\({\rm{ For n = 20, the following are histogram and normal probability plot: }}\)

04

Determining the sample size

\({\rm{ For n = 30, the following are histogram and normal probability plot: }}\)

05

Determining the sample size

\({\rm{ For n = 50, the following are histogram and normal probability plot: }}\)

06

Determining the sample size

\({\rm{ For n = 300, the following are histogram and normal probability plot: }}\)

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Most popular questions from this chapter

Question: The number of customers waiting for gift-wrap service at a department store is an rv X with possible values \({\rm{0,1,2,3,4}}\)and corresponding probabilities \({\rm{.1,}}{\rm{.2,}}{\rm{.3,}}{\rm{.25,}}{\rm{.15}}{\rm{.}}\)A randomly selected customer will have \({\rm{1,2}}\),or \({\rm{3}}\) packages for wrapping with probabilities \({\rm{.6,}}{\rm{.3,}}\)and \({\rm{.1,}}\)respectively. Let \({\rm{Y = }}\)the total number of packages to be wrapped for the customers waiting in line (assume that the number of packages submitted by one customer is independent of the number submitted by any other customer).

a. Determine \({\rm{P(X = 3,Y = 3)}}\), i.e., \({\rm{P(3,3)}}\).

b. Determine \({\rm{p(4,11)}}\).

Answer the following questions:

a. Given that\({\rm{X = 1}}\), determine the conditional pmf of \({\rm{Y}}\)-i.e., \({{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(0}}\mid {\rm{1),}}{{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(1}}\mid {\rm{1)}}\), and\({{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(2}}\mid {\rm{1)}}\).

b. Given that two houses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island?

c. Use the result of part (b) to calculate the conditional probability\({\rm{P(Y拢

1}}\mid {\rm{X = 2)}}\).

d. Given that two houses are in use at the full-service island, what is the conditional pmf of the number in use at the self-service island?

A health-food store stocks two different brands of a certain type of grain. Let \(X = \)the amount (lb) of brand A on hand and \(Y = \)the amount of brand B on hand. Suppose the joint pdf of X and Y is

\(f(x,y) = \left\{ {\begin{array}{*{20}{c}}{kxy\;\;\;\;\;\;\;x \ge 0,\;y \ge 0,\;20 \le x + y \le 30}\\{0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise}\end{array}} \right\}\)

a. Draw the region of positive density and determine the value of k.

b. Are X and Y independent? Answer by first deriving the marginal pdf of each variable.

c. Compute \(P\left( {X + Y \le 25} \right)\).

d. What is the expected total amount of this grain on hand?

e. Compute \(Cov\left( {X, Y} \right)\)and\(Corr\left( {X, Y} \right)\).

f. What is the variance of the total amount of grain on hand?

Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv鈥檚 with expected values \({\mu _1}, {\mu _2}, and {\mu _3}\)and variances \(\sigma _1^2 , \sigma _2^2, and \sigma _3^2 \), respectively. a. If \(\mu = {\mu _2} = {\mu _3} = 60\)and\(\sigma _1^2 = \sigma _2^2 = \sigma _3^2 = 15\), calculate \(P\left( {{T_0} \le 200} \right)\)and\(P\left( {150 \le {T_0} \le 200} \right)\)? b. Using the \(\mu 's and \sigma 's\)given in part (a), calculate both \(P\left( {55 \le X} \right)\)and \(P\left( {58 \le X \le 62} \right)\).c. Using the \(\mu 's and \sigma 's\)given in part (a), calculate and interpret\(P\left( { - 10 \le {X_1} - .5{X_2} - .5{X_3} \le 5} \right)\). d. If\({\mu _1} = 40, {\mu _1} = 50, {\mu _1} = 60,\),\( \sigma _1^2 = 10, \sigma _2^2 = 12, and \sigma _3^2 = 14\) calculate \(P\left( {{X_1} + {X_2} + {X_3} \le 160} \right)\)and also \(P\left( {{X_1} + {X_2} \ge 2{X_3}} \right).\)

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.

See all solutions

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