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Chapter 5: Q57E (page 238)

Suppose the distribution of the time X (in hours) spent by students at a certain university on a particular project is gamma with parameters \(\alpha = 50\)and\(\beta = 2\). Because a is large, it can be shown that X has an approximately normal distribution. Use this fact to compute the approximate probability that a randomly selected student spends at most \(125\) hours on the project.

Short Answer

Expert verified

The approximate probability that a randomly selected student spends at most 125 hours on the project is \(96.16\% \).

Step by step solution

01

Definition of Probability

The likelihood of an event happening is defined by probability. We may be required to forecast the outcome of an event in a variety of real-life scenarios. The outcomes of an event may or may not be certain. In these instances, we state that the event has a chance of happening or not happening.

02

Calculation for the determination of the probability.

Given:

\(\begin{array}{l}\alpha = 50\\\beta = 2\end{array}\)

The mean of a gamma distribution is the product of its variables\(\alpha \;and\;\beta \):

\(\mu = E(X) = \alpha \beta = 50(2) = 100\)

The variance of a gamma distribution is the product of its variables squared\(\alpha \;and\;\beta \):

\({\sigma ^2} = V(X) = \alpha {\beta ^2} = 50{(2)^2} = 50(4) = 200\)

The standard deviation is the square root of the variance:

\(\sigma = \sqrt {200} = 10\sqrt 2 \approx 14.1421\)

03

Further calculation for the determination of the probability.

Given is that it is appropriate to approximate the gamma distribution with the normal distribution. We will approximate the gamma distribution with the normal distribution of mean \(\mu = 100\)and standard deviation

\(\sigma = 10\sqrt 2 \approx 14.1421\)

Note: No continuity correction is required, since gamma distribution is a continuous function and the normal distribution is also a continuous function.

The z-score is the value decreased by the mean and divided by the standard deviation.

\(z = \frac{{x - np}}{{\sqrt {np(1 - p)} }} = \frac{{125 - 100}}{{10\sqrt 2 }} \approx 1.77\)

Determine the corresponding normal probability using table.

\(P(X \le 125) = P(Z < 1.77) = 0.9616 = 96.16\% \)

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Most popular questions from this chapter

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

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\({{\rm{\sigma }}^{\rm{2}}}\).

Let \({\rm{A}}\)denote the percentage of one constituent in a randomly selected rock specimen, and let \({\rm{B}}\)denote the percentage of a second constituent in that same specimen. Suppose \({\rm{D}}\)and \({\rm{E}}\)are measurement errors in determining the values of \({\rm{A}}\)and \({\rm{B}}\)so that measured values are \({\rm{X = A + D}}\)and\({\rm{Y = B + E}}\), respectively. Assume that measurement errors are independent of one another and of actual values.

a. Show that

\({\rm{Corr(X,Y) = Corr(A,B) \times }}\sqrt {{\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right)} {\rm{ \times }}\sqrt {{\rm{Corr}}\left( {{{\rm{Y}}_{\rm{1}}}{\rm{,}}{{\rm{Y}}_{\rm{2}}}} \right)} \)

where \({{\rm{X}}_{\rm{1}}}\)and \({{\rm{X}}_{\rm{2}}}\)are replicate measurements on the value of\({\rm{A}}\), and \({{\rm{Y}}_{\rm{1}}}\)and \({{\rm{Y}}_{\rm{2}}}\)are defined analogously with respect to\({\rm{B}}\). What effect does the presence of measurement error have on the correlation?

b. What is the maximum value of \({\rm{Corr(X,Y)}}\)when \({\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.8100}}\)and \({\rm{Corr}}\left( {{{\rm{Y}}_{\rm{1}}}{\rm{,}}{{\rm{Y}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.9025?}}\)Is this disturbing?

Suppose the proportion of rural voters in a certain state who favor a particular gubernatorial candidate is\(.{\bf{45}}\)and the proportion of suburban and urban voters favouring the candidate is\(.{\bf{60}}\). If a sample of\({\bf{200}}\)rural voters and\({\bf{300}}\)urban and suburban voters is obtained, what is the approximate probability that at least\(\;{\bf{250}}\)of these voters favour this candidate?

Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv鈥檚 with expected values \({\mu _1}, {\mu _2}, and {\mu _3}\)and variances \(\sigma _1^2 , \sigma _2^2, and \sigma _3^2 \), respectively. a. If \(\mu = {\mu _2} = {\mu _3} = 60\)and\(\sigma _1^2 = \sigma _2^2 = \sigma _3^2 = 15\), calculate \(P\left( {{T_0} \le 200} \right)\)and\(P\left( {150 \le {T_0} \le 200} \right)\)? b. Using the \(\mu 's and \sigma 's\)given in part (a), calculate both \(P\left( {55 \le X} \right)\)and \(P\left( {58 \le X \le 62} \right)\).c. Using the \(\mu 's and \sigma 's\)given in part (a), calculate and interpret\(P\left( { - 10 \le {X_1} - .5{X_2} - .5{X_3} \le 5} \right)\). d. If\({\mu _1} = 40, {\mu _1} = 50, {\mu _1} = 60,\),\( \sigma _1^2 = 10, \sigma _2^2 = 12, and \sigma _3^2 = 14\) calculate \(P\left( {{X_1} + {X_2} + {X_3} \le 160} \right)\)and also \(P\left( {{X_1} + {X_2} \ge 2{X_3}} \right).\)
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