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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

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?

Let \({\rm{X}}\) be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of \({\rm{X}}\) is as follows:

a. Consider a random sample of size \({\rm{n = 2}}\) (two customers), and let \({\rm{\bar X}}\) be the sample mean number of packages shipped. Obtain the probability distribution of\({\rm{\bar X}}\).

b. Refer to part (a) and calculate\({\rm{P(\bar X\pounds2}}{\rm{.5)}}\).

c. Again consider a random sample of size\({\rm{n = 2}}\), but now focus on the statistic \({\rm{R = }}\) the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of\({\rm{R}}\). (Hint: Calculate the value of \({\rm{R}}\) for each outcome and use the probabilities from part (a).)

d. If a random sample of size \({\rm{n = 4}}\) is selected, what is \({\rm{P(\bar X\pounds1}}{\rm{.5)}}\) ? (Hint: You should not have to list all possible outcomes, only those for which\({\rm{\bar x\pounds1}}{\rm{.5}}\).)

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 Weibull with \({\rm{\alpha = 2}}\) and\({\rm{\beta = 5}}\), as in Example\({\rm{5}}{\rm{.20}}\).[A1] Consider the four sample sizes, and\({\rm{30}}\), 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?

You have two lightbulbs for a particular lamp. Let\({\rm{X = }}\)the lifetime of the first bulb and\({\rm{Y = }}\)the lifetime of the second bulb (both in\({\rm{1000}}\)s of hours). Suppose that\({\rm{X}}\)and\({\rm{Y}}\)are independent and that each has an exponential distribution with parameter\({\rm{\lambda = 1}}\).

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

b. What is the probability that each bulb lasts at most\({\rm{1000}}\)hours (i.e.,\({\rm{X拢1}}\)and\({\rm{Y拢1)}}\)?

c. What is the probability that the total lifetime of the two bulbs is at most\({\rm{2}}\)? (Hint: Draw a picture of the region before integrating.)

d. What is the probability that the total lifetime is between\({\rm{1}}\)and\({\rm{2}}\)?

Garbage trucks entering a particular waste-management facility are weighed prior to offloading their contents. Let \({\rm{X = }}\)the total processing time for a randomly selected truck at this facility (waiting, weighing, and offloading). The article "Estimating Waste Transfer Station Delays Using GPS" (Waste Mgmt., \({\rm{2008: 1742 - 1750}}\)) suggests the plausibility of a normal distribution with mean \({\rm{13\;min}}\)and standard deviation \({\rm{4\;min}}\)for\({\rm{X}}\). Assume that this is in fact the correct distribution.

a. What is the probability that a single truck's processing time is between \({\rm{12}}\) and \({\rm{15\;min}}\)?

b. Consider a random sample of \({\rm{16}}\) trucks. What is the probability that the sample mean processing time is between \({\rm{12}}\) and\({\rm{15\;min}}\)?

c. Why is the probability in (b) much larger than the probability in (a)?

d. What is the probability that the sample mean processing time for a random sample of \({\rm{16}}\) trucks will be at least\({\rm{20\;min}}\)?
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