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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

According to the article 鈥淩eliability Evaluation of Hard Disk Drive Failures Based on Counting Processes鈥� (Reliability Engr. and System Safety,\({\bf{2013}}:{\rm{ }}{\bf{110}}--{\bf{118}}\)), particles accumulating on a disk drive come from two sources, one external and the other internal. The article proposed a model in which the internal source contains a number of loose particles W having a Poisson distribution with mean value of m; when a loose particle releases, it immediately enters the drive, and the release times are independent and identically distributed with cumulative distribution function G(t). Let X denote the number of loose particles not yet released at a particular time t. Show that X has a Poisson distribution with parameter\(\mu \left( {1 - G\left( t \right)} \right)\). (Hint: Let Y denote the number of particles accumulated on the drive from the internal source by time t so that\(X + Y = W\). Obtain an expression for\(P\left( {X = x, Y = y} \right)\)and then sum over y.)

Let \({\rm{X}}\)and \({\rm{Y}}\)be independent standard normal random variables, and define a new rv by \({\rm{U = }}{\rm{.6X + }}{\rm{.8Y}}\).

a. \({\rm{Determine\;Corr(X,U)}}\)

b. How would you alter \({\rm{U}}\)to obtain \({\rm{Corr(X,U) = \rho }}\)for a specified value of \({\rm{\rho ?}}\)

Two different professors have just submitted final exams for duplication. Let \({\rm{X}}\) denote the number of typographical errors on the first professor鈥檚 exam and \({\rm{Y}}\) denote the number of such errors on the second exam. Suppose \({\rm{X}}\) has a Poisson distribution with parameter \({{\rm{\mu }}_{\rm{1}}}\), \({\rm{Y}}\) has a Poisson distribution with parameter \({{\rm{\mu }}_{\rm{2}}}\), and \({\rm{X}}\) and \({\rm{Y}}\) are independent.

a. What is the joint pmf of \({\rm{X}}\) and\({\rm{Y}}\)?

b. What is the probability that at most one error is made on both exams combined?

c. Obtain a general expression for the probability that the total number of errors in the two exams is m (where \({\rm{m}}\) is a nonnegative integer). (Hint: \({\rm{A = }}\left\{ {\left( {{\rm{x,y}}} \right){\rm{:x + y = m}}} \right\}{\rm{ = }}\left\{ {\left( {{\rm{m,0}}} \right)\left( {{\rm{m - 1,1}}} \right){\rm{,}}.....{\rm{(1,m - 1),(0,m)}}} \right\}\)Now sum the joint pmf over \({\rm{(x,y)}} \in {\rm{A}}\)and use the binomial theorem, which says that

\({\rm{P(X + Y = m)}}\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\sum\limits_{{\rm{k = 0}}}^{\rm{m}} {\left( {\begin{array}{*{20}{c}}{\rm{m}}\\{\rm{k}}\end{array}} \right){{\rm{a}}^{\rm{k}}}{{\rm{b}}^{{\rm{m - k}}}}{\rm{ = }}\left( {{\rm{a + b}}} \right)} ^{\rm{m}}}\)

An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let \({\rm{X}}\)=the \({\rm{x}}\) coordinate of the point selected and \({\rm{Y}}\)=the \({\rm{y}}\) coordinate of the point selected. If the circle is centered at \({\rm{(0,0)}}\)and has radius \({\rm{R}}\), then the joint pdf of \({\rm{X}}\)and \({\rm{Y}}\) is

\({\rm{f(x,y) = }}\left\{ {\begin{array}{*{20}{c}}{\frac{{\rm{1}}}{{{\rm{\pi }}{{\rm{R}}^{\rm{2}}}}}}&{{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{拢}}{{\rm{R}}^{\rm{2}}}}\\{\rm{0}}&{{\rm{\;otherwise\;}}}\end{array}} \right.\)

a. What is the probability that the selected point is within \(\frac{{\rm{R}}}{{\rm{2}}}\)of the center of the circular region? (Hint: Draw a picture of the region of positive density \({\rm{D}}\). Because \({\rm{f}}\)(\({\rm{x}}\), \({\rm{y}}\)) is constant on \({\rm{D}}\), computing a probability reduces to computing an area.)

b. What is the probability that both \({\rm{X and Y}}\)differ from 0 by at most\(\frac{{\rm{R}}}{{\rm{2}}}\)?

c. Answer part (b) for\(\frac{{\rm{R}}}{{\sqrt {\rm{2}} }}\)replacing\(\frac{{\rm{R}}}{{\rm{2}}}\)

d. What is the marginal pdf of \({\rm{X}}\)? Of \({\rm{Y}}\)? Are \({\rm{X and Y}}\)independent?

A particular brand of dishwasher soap is sold in three sizes: \({\rm{25oz,40oz}}\), and\({\rm{65oz}}\). Twenty percent of all purchasers select a\({\rm{25 - 0z}}\)box,\({\rm{50\% }}\)select a\({\rm{40 - 0z}}\)box, and the remaining\({\rm{30\% }}\)choose a\({\rm{65}}\)-oz box. Let\({{\rm{X}}_{\rm{1}}}\)and\({{\rm{X}}_{\rm{2}}}\)denote the package sizes selected by two independently selected purchasers.

a. Determine the sampling distribution of\({\rm{\bar X}}\), calculate\({\rm{E(\bar X)}}\), and compare to\({\rm{\mu }}\).

b. Determine the sampling distribution of the sample variance\({{\rm{S}}^{\rm{2}}}\), calculate\({\rm{E}}\left( {{{\rm{S}}^{\rm{2}}}} \right)\), and compare to\({{\rm{\sigma }}^{\rm{2}}}\).
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