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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

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?

In an area having sandy soil,\({\rm{50}}\)small trees of a certain type were planted, and another \({\rm{50}}\) trees were planted in an area having clay soil. Let \({\rm{X = }}\) the number of trees planted in sandy soil that survive \({\rm{1}}\) year and \({\rm{Y = }}\)the number of trees planted in clay soil that survive \({\rm{1}}\) year. If the probability that a tree planted in sandy soil will survive \({\rm{1}}\)year is \({\rm{.7}}\)and the probability of \({\rm{1}}\)-year survival in clay soil is \({\rm{.6}}\), compute an approximation to \({\rm{P( - 5拢 X - Y拢 5)}}\) (do not bother with the continuity correction).

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 \({\rm{10}}\) ?

Annie and Alvie have agreed to meet between\({\rm{5:00 P}}{\rm{.M}}\). and\({\rm{6:00 P}}{\rm{.M}}\). for dinner at a local health-food restaurant. Let\({\rm{X = }}\)Annie's arrival time and\({\rm{Y = }}\)Alvie's arrival time. Suppose\({\rm{X}}\)and\({\rm{Y}}\)are independent with each uniformly distributed on the interval\({\rm{(5,6)}}\).

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

b. What is the probability that they both arrive between\({\rm{5:15}}\)and\({\rm{5:45}}\)?

c. If the first one to arrive will wait only \({\rm{10\;min}}\) before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? (Hint: The event of interest is\({\rm{A = \{(x,y):|x - y|拢1/6\}}}\).)

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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