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Chapter 5: Q52E (page 237)

The lifetime of a certain type of battery is normally distributed with mean value \({\rm{10}}\)hours and standard deviation \({\rm{1}}\)hour. There are four batteries in a package. What lifetime value is such that the total lifetime of all batteries in a package exceeds that value for only \({\rm{5\% }}\)of all packages?

Short Answer

Expert verified

\({\rm{43}}{\rm{.29}}\) is the total lifetime of all batteries in a package exceeds that value for only \({\rm{5\% }}\)of all packages

Step by step solution

01

Definition of standard deviation

The square root of the variance is the standard deviation of a random variable, sample, statistical population, data collection, or probability distribution. It is less resilient in practice than the average absolute deviation, but it is algebraically easier.

02

Determining the total lifetime of all batteries in a package exceeds that value for only \({\rm{5\% }}\) of all packages

The random variable in question has a normal distribution with a mean of 10 and a standard deviation of 1. Examine the total of four random variables of this type.

\({{\rm{T}}_{\rm{0}}}{\rm{ = }}{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}{\rm{ + }}{{\rm{X}}_{\rm{4}}}{\rm{,}}\)

Each random variable represents one battery. The standard deviation is and the mean value of \({{\rm{T}}_{\rm{0}}}\) is

\({{\rm{\mu }}_{{{\rm{T}}_{\rm{0}}}}}{\rm{ = n \times \mu = 4 \times 10 = 40}}\)

The standard deviation is

\({{\rm{\sigma }}_{{{\rm{T}}_{\rm{0}}}}}{\rm{ = }}\sqrt {\rm{n}} {\rm{ \times \sigma = }}\sqrt {\rm{4}} {\rm{ \times 1 = 2}}{\rm{.}}\)

The \({\rm{9}}{{\rm{5}}^{{\rm{th\;}}}}\) percentile of random variable \({{\rm{T}}_{\rm{0}}}\)is the desired value. The \({{\rm{z}}_{{\rm{0}}{\rm{.05}}}}\) percentile of a standard normal distribution is \({\rm{9}}{{\rm{5}}^{{\rm{th\;}}}}\)

\({{\rm{z}}_{{\rm{0}}{\rm{.05}}}}{\rm{ = 1}}{\rm{.645}}{\rm{.\;}}\)

Consequently, the values \({\rm{9}}{{\rm{5}}^{{\rm{th\;}}}}\) random variable percentile \({{\rm{T}}_{\rm{0}}}\)

\({{\rm{\sigma }}_{{{\rm{T}}_{\rm{0}}}}}{\rm{ + }}{{\rm{z}}_{{\rm{0}}{\rm{.05}}}}{\rm{ \times }}{{\rm{\mu }}_{{{\rm{T}}_{\rm{0}}}}}{\rm{ = 40 + 1}}{\rm{.645 \times 2 = 43}}{\rm{.29}}{\rm{.}}\)

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Most popular questions from this chapter

Let \({\rm{X}}\)and \({\rm{Y}}\)be independent standard normal random variables, and define a new rv by \({\rm{U = }}{\rm{.6X + }}{\rm{.8Y}}\).

a. \({\rm{Determine\;Corr(X,U)}}\)

b. How would you alter \({\rm{U}}\)to obtain \({\rm{Corr(X,U) = \rho }}\)for a specified value of \({\rm{\rho ?}}\)

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

a. For \({\rm{f}}\)(\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}\)) as given in Example \({\rm{5}}{\rm{.10}}\), compute the joint marginal density function of \({{\rm{X}}_{\rm{1}}}{\rm{and}}{{\rm{X}}_{\rm{3}}}\)alone (by integrating over x2).

b. What is the probability that rocks of types \({\rm{1 and 3}}\)together make up at most \({\rm{50\% }}\)of the sample? (Hint: Use the result of part (a).)

c. Compute the marginal pdf of \({{\rm{X}}_{\rm{1}}}\) alone. (Hint: Use the result of part (a).)

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?
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