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Chapter 5: Q44E (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 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?

Short Answer

Expert verified

The distribution for sample size \({\rm{30}}\) looks to be more normal.

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

We discovered that as sample size grows, the distribution tends to normalize. For example, the distribution for sample size \({\rm{30}}\) looks to be more normal.

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Most popular questions from this chapter

A surveyor wishes to lay out a square region with each side having length\({\rm{L}}\). However, because of a measurement error, he instead lays out a rectangle in which the north-south sides both have length \({\rm{X}}\) and the east-west sides both have length\({\rm{Y}}\). Suppose that \({\rm{X}}\) and \({\rm{Y}}\) are independent and that each is uniformly distributed on the interval \({\rm{(L - A,L + A)}}\) (where \({\rm{0 < A < L}}\) ). What is the expected area of the resulting rectangle?

Suppose a randomly chosen individual's verbal score \({\rm{X}}\)and quantitative score \({\rm{Y}}\)on a nationally administered aptitude examination have a joint pdf

\({\rm{f(x,y) = }}\left\{ {\begin{array}{*{20}{c}}{\frac{{\rm{2}}}{{\rm{5}}}{\rm{(2x + 3y)}}}&{{\rm{0拢 x拢 1,0拢 y拢 1}}}\\{\rm{0}}&{{\rm{ otherwise }}}\end{array}} \right.\)

You are asked to provide a prediction \({\rm{t}}\)of the individual's total score\({\rm{X + Y}}\). The error of prediction is the mean squared error\({\rm{E}}\left( {{{{\rm{(X + Y - t)}}}^{\rm{2}}}} \right)\). What value of \({\rm{t}}\)minimizes the error of prediction?

A more accurate approximation to \({\rm{E}}\left( {{\rm{h}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}} \right)} \right)\) in Exercise 95 is

\(h\left( {{\mu _1}, \ldots ,{\mu _n}} \right) + \frac{1}{2}\sigma _1^2\left( {\frac{{{\partial ^2}h}}{{\partial x_1^2}}} \right) + \cdots + \frac{1}{2}\sigma _n^2\left( {\frac{{{\partial ^2}h}}{{\partial x_n^2}}} \right)\)

Compute this for \({\rm{Y = h}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}{\rm{,}}{{\rm{X}}_{\rm{4}}}} \right)\)given in Exercise 93 , and compare it to the leading term \({\rm{h}}\left( {{{\rm{\mu }}_{\rm{1}}}{\rm{, \ldots ,}}{{\rm{\mu }}_{\rm{n}}}} \right)\).

The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter \({\rm{ \mu = 50}}\).

a. Calculate the approximate probability that between \({\rm{35 and 70 }}\)tickets are given out on a particular day.

b. Calculate the approximate probability that the total number of tickets given out during a \({\rm{5 - }}\)day week is between \({\rm{225 and 275}}\)

c. Use software to obtain the exact probabilities in (a) and (b) and compare to their approximations.

Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of \({\rm{1}}{\rm{.2}}{\rm{.}}\)

a. If the distribution is normal, what is the probability that the sample mean hardness for a random sample of \({\rm{9}}\) pins is at least \({\rm{51}}\)?

b. Without assuming population normality, what is the (approximate) probability that the sample mean hardness for a random sample of \({\rm{40 }}\) pins is at least \({\rm{51}}\)?

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