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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

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\({\rm{2}}\)?

A restaurant serves three fixed-price dinners costing \({\rm{12, 15and 20}}\). For a randomly selected couple dining at this restaurant, let \({\rm{X = }}\)the cost of the man鈥檚 dinner and \({\rm{Y = }}\)the cost of the woman鈥檚 dinner. The joint \({\rm{pmf's}}\) of \({\rm{X and Y }}\)is given in the following table:

a. Compute the marginal \({\rm{pmf's}}\)of \({\rm{X and Y }}\)

b. What is the probability that the man鈥檚 and the woman鈥檚 dinner cost at most \({\rm{15}}\)each?

c. Are \({\rm{X and Y }}\)independent? Justify your answer.

d. What is the expected total cost of the dinner for the two people?

e. Suppose that when a couple opens fortune cookies at the conclusion of the meal, they find the message 鈥淵ou will receive as a refund the difference between the cost of the more expensive and the less expensive meal that you have chosen.鈥� How much would the restaurant expect to refund?

Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean \({\rm{2}}{\rm{.65 }}\)and standard deviation \({\rm{.85}}\) (suggested in 鈥淢odeling Sediment and Water Column Interactions for Hydrophobic Pollutants,鈥� Water Research, \({\rm{1984: 1164 - 1174 }}\)).

a. If a random sample of \({\rm{25}}\)specimens is selected, what is the probability that the sample average sediment density is at most \({\rm{3}}{\rm{.00 }}\)? Between \({\rm{2}}{\rm{.65 }}\)and \({\rm{3}}{\rm{.00 }}\)?

b. How large a sample size would be required to ensure that the first probability in part (a) is at least \({\rm{.99}}\)?

A stockroom currently has \({\rm{30}}\) components of a certain type, of which \({\rm{8}}\) were provided by supplier \({\rm{1,10}}\)by supplier \({\rm{2}}\) , and \({\rm{12}}\) by supplier \({\rm{3}}\). Six of these are to be randomly selected for a particular assembly. Let \({\rm{X = }}\) the number of supplier l's components selected, \({\rm{Y = }}\) the number of supplier \({\rm{2}}\) 's components selected, and \({\rm{p(x,y)}}\) denote the joint pmf of \({\rm{X}}\) and\({\rm{Y}}\).

a. What is \({\rm{p(3,2)}}\) ? (Hint: Each sample of size \({\rm{6}}\) is equally likely to be selected. Therefore, \({\rm{p(3,2) = }}\) (number of outcomes with \({\rm{X = 3}}\) and \({\rm{Y = 2)/(}}\) total number of outcomes). Now use the product rule for counting to obtain the numerator and denominator.)

b. Using the logic of part (a), obtain \({\rm{p(x,y}}\) ). (This can be thought of as a multivariate hypergeometric distribution-sampling without replacement from a finite population consisting of more than two categories.)

Answer

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?
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