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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

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?

A company maintains three offices in a certain region, each staffed by two employees. Information concerning yearly salaries (\({\rm{1000}}\)s of dollars) is as follows:

\(\begin{array}{*{20}{c}}{{\rm{ Office }}}&{\rm{1}}&{\rm{1}}&{\rm{2}}&{\rm{2}}&{\rm{3}}&{\rm{3}}\\{{\rm{ Employee }}}&{\rm{1}}&{\rm{2}}&{\rm{3}}&{\rm{4}}&{\rm{5}}&{\rm{6}}\\{{\rm{ Salary }}}&{{\rm{29}}{\rm{.7}}}&{{\rm{33}}{\rm{.6}}}&{{\rm{30}}{\rm{.2}}}&{{\rm{33}}{\rm{.6}}}&{{\rm{25}}{\rm{.8}}}&{{\rm{29}}{\rm{.7}}}\end{array}\)

a. Suppose two of these employees are randomly selected from among the six (without replacement). Determine the sampling distribution of the sample mean salary\({\rm{\bar X}}\).

b. Suppose one of the three offices is randomly selected. Let\({{\rm{X}}_{\rm{1}}}\)and\({{\rm{X}}_{\rm{2}}}\)denote the salaries of the two employees. Determine the sampling distribution of\({\rm{\bar X}}\).

c. How does \({\rm{E(\bar X)}}\) from parts (a) and (b) compare to the population mean salary\({\rm{\mu }}\)?

The difference between the number of customers in line at the express checkout and the number in line at the super-express checkout is\({{\rm{X}}_{\rm{1}}}{\rm{ - }}{{\rm{X}}_{\rm{2}}}\). Calculate the expected difference.

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 \({\rm{.99}}\)?

Return to the situation described in Exercise \({\rm{3}}\).

a. Determine the marginal pmf of \({{\rm{X}}_{\rm{1}}}\), and then calculate the expected number of customers in line at the express checkout.

b. Determine the marginal pmf of \({{\rm{X}}_{\rm{2}}}\).

c. By inspection of the probabilities \({\rm{P(}}{{\rm{X}}_{\rm{1}}}{\rm{ = 4),P(}}{{\rm{X}}_{\rm{2}}}{\rm{ = 0),}}\) and \({\rm{P(}}{{\rm{X}}_{\rm{1}}}{\rm{ = 4,}}{{\rm{X}}_{\rm{2}}}{\rm{ = 0),}}\) are \({{\rm{X}}_{\rm{1}}}\) and \({{\rm{X}}_{\rm{2}}}\) independent random variables? Explain
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