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Chapter 5: Q49E (page 237)

There are \({\rm{40}}\) students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of \({\rm{6}}\)min and a standard deviation of \({\rm{6}}\)min.

a. If grading times are independent and the instructor begins grading at \({\rm{6:50}}\) p.m. and grades continuously, what is the (approximate) probability that he is through grading before the \({\rm{11:00}}\) p.m. TV news begins?

b. If the sports report begins at \({\rm{11:10,}}\) what is the probability that he misses part of the report if he waits until grading is done before turning on the TV?

Short Answer

Expert verified

a) \(P\left( {{T_0}拢250} \right) = 0.6026\)

b) \({\rm{P}}\left( {{{\rm{T}}_{\rm{0}}}{\rm{ > 260}}} \right){\rm{ = 0}}{\rm{.2981}}\)

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 (approximate) probability that he is through grading before the \({\rm{11:00}}\) p.m. TV news begins

There is a total of 250 minutes between 6:50 PM and \({\rm{11:00PM}}\). The sum of the 40 random variables provided in the exercise can be used to indicate total grading time (mean 6 minutes, standard deviation 6 minutes). Define it as follows:

\({{\rm{T}}_{\rm{0}}}{\rm{ = }}{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + \ldots + }}{{\rm{X}}_{{\rm{40}}}}{\rm{.}}\)

This random variable's mean value is

\({{\rm{\mu }}_{{{\rm{T}}_{\rm{0}}}}}{\rm{ = n \times \mu = 40 \times 6 = 240,}}\)

as well as the standard deviation

\({{\rm{\sigma }}_{{{\rm{T}}_{\rm{0}}}}}{\rm{ = }}\sqrt {\rm{n}} {\rm{ \times \sigma = }}\sqrt {{\rm{40}}} {\rm{ \times 6 = 37}}{\rm{.95}}{\rm{.}}\)

It is simple to determine the requested probability by determining the mean and standard deviation of the random variable \({{\rm{T}}_{\rm{0}}}\) as follows:

\(\begin{array}{*{20}{c}}{{\rm{P}}\left( {{{\rm{T}}_{\rm{0}}}{\rm{拢 250}}} \right)}&{{\rm{ = P}}\left( {\frac{{{{\rm{T}}_{\rm{0}}}{\rm{ - }}{{\rm{\mu }}_{{{\rm{T}}_{\rm{0}}}}}}}{{{{\rm{\sigma }}_{{{\rm{T}}_{\rm{0}}}}}}}{\rm{拢 }}\frac{{{\rm{250 - 240}}}}{{{\rm{37}}{\rm{.95}}}}} \right){\rm{ = P(Z拢 0}}{\rm{.26)}}}\\{}&{\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{0}}{\rm{.6026}}}\end{array}\)

(\({\rm{1}}\)): from the appendix's normal probability table. Software can also be used to calculate the likelihood.

03

Determining the probability that he misses part of the report if he waits until grading is done before turning on the TV

There are \({\rm{260}}\) minutes left until the sports report starts. The following statement is correct:

\(\begin{array}{*{20}{c}}{{\rm{P}}\left( {{{\rm{T}}_{\rm{0}}}{\rm{ > 260}}} \right){\rm{ = P}}\left( {\frac{{{{\rm{T}}_{\rm{0}}}{\rm{ - }}{{\rm{\mu }}_{{{\rm{T}}_{\rm{0}}}}}}}{{{{\rm{\sigma }}_{{{\rm{T}}_{\rm{0}}}}}}}{\rm{ > }}\frac{{{\rm{260 - 240}}}}{{{\rm{37}}{\rm{.95}}}}} \right){\rm{ = P(Z > 0}}{\rm{.53)}}}\\{\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{0}}{\rm{.2981}}{\rm{.}}}\end{array}\)

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Most popular questions from this chapter

If two loads are applied to a cantilever beam as shown in the accompanying drawing, the bending moment at \({\rm{0}}\) due to the loads is \({{\rm{a}}_{\rm{1}}}{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{a}}_{\rm{2}}}{{\rm{X}}_{\rm{2}}}{\rm{ }}\)

a. Suppose that \({{\rm{X}}_{\rm{1}}}{\rm{ and }}{{\rm{X}}_{\rm{2}}}{\rm{ }}\)are independent rv鈥檚 with means \({\rm{2 and 4kip}}\), respectively, and standard deviations \({\rm{.5}}\) and \({\rm{1}}{\rm{.0kip}}\), respectively. If \({{\rm{a}}_{\rm{1}}}{\rm{ = 5ft and }}{{\rm{a}}_{\rm{2}}}{\rm{ = 10ft }}\), what is the expected bending moment and what is the standard deviation of the bending moment?

b. If \({{\rm{X}}_{\rm{1}}}{\rm{ and }}{{\rm{X}}_{\rm{2}}}{\rm{ }}\)are normally distributed, what is the probability that the bending moment will exceed 75 kip-ft?

c. Suppose the positions of the two loads are random variables. Denoting them by \({{\rm{A}}_{\rm{1}}}{\rm{ and }}{{\rm{A}}_{\rm{2}}}{\rm{ }}\), assume that these variables have means of \({\rm{5 and 10ft }}\), respectively, that each has a standard deviation of \({\rm{.5}}\), and that all are independent of one another. What is the expected moment now?

d. For the situation of part (c), what is the variance of the bending moment?

e. If the situation is as described in part (a) except that \({\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = 0}}{\rm{.5}}\) (so that the two loads are not independent), what is the variance of the bending moment?

A shipping company handles containers in three different sizes:\(\left( 1 \right)\;27f{t^3}\;\left( {3 \times 3 \times 3} \right)\)\(\left( 2 \right) 125 f{t^3}, and \left( 3 \right)\;512 f{t^3}\). Let \({X_i}\left( {i = \;1, 2, 3} \right)\)denote the number of type i containers shipped during a given week. With \({\mu _i} = E\left( {{X_i}} \right)\)and\(\sigma _i^2 = V\left( {{X_i}} \right)\), suppose that the mean values and standard deviations are as follows:

\(\begin{array}{l}{\mu _1} = 200 {\mu _2} = 250 {\mu _3} = 100 \\{\sigma _1} = 10 {\sigma _2} = \,12 {\sigma _3} = 8\end{array}\)

a. Assuming that \({X_1}, {X_2}, {X_3}\)are independent, calculate the expected value and variance of the total volume shipped. (Hint:\(Volume = 27{X_1} + 125{X_2} + 512{X_3}\).)

b. Would your calculations necessarily be correct if \({X_i} 's\)were not independent?Explain.

Suppose the expected tensile strength of type-A steel is \({\rm{105ksi}}\)and the standard deviation of tensile strength is \({\rm{8ksi}}\). For type-B steel, suppose the expected tensile strength and standard deviation of tensile strength are \({\rm{100ksi}}\)and \({\rm{6ksi}}\), respectively. Let \({\rm{\bar X = }}\)the sample average tensile strength of a random sample of \({\rm{40}}\) type-A specimens, and let \({\rm{\bar Y = }}\)the sample average tensile strength of a random sample of \({\rm{35}}\)type-B specimens.

a. What is the approximate distribution of \({\rm{\bar X ? of \bar Y?}}\)

b. What is the approximate distribution of \({\rm{\bar X - \bar Y}}\)? Justify your answer.

c. Calculate (approximately) \(P( - 1拢\bar X - \bar Y拢1)\)

d. Calculate. If you actually observed , would you doubt that \({{\rm{\mu }}_{\rm{1}}}{\rm{ - }}{{\rm{\mu }}_{\rm{2}}}{\rm{ = 5?}}\)

Consider a system consisting of three components as pictured. The system will continue to function as long as the first component functions and either component \({\rm{2}}\) or component \({\rm{3}}\)functions. Let \({{\rm{X}}_{{\rm{1,}}}}{{\rm{X}}_{\rm{2}}}\), and \({{\rm{X}}_{\rm{3}}}\) denote the lifetimes of components \({\rm{1}}\), \({\rm{2}}\), and \({\rm{3}}\), respectively. Suppose the \({{\rm{X}}_{\rm{i}}}\) 鈥檚 are independent of one another and each \({{\rm{X}}_{\rm{i}}}\) has an exponential distribution with parameter \({\rm{\lambda }}\).

a. Let \({\rm{Y}}\) denote the system lifetime. Obtain the cumulative distribution function of \({\rm{Y}}\)and differentiate to obtain the pdf. (Hint: \({{\rm{F}}_{\left( {\rm{Y}} \right)}}{\rm{P}}\left\{ {{\rm{Y}} \le {\rm{y}}} \right\}\); express the event \(\left\{ {{\rm{Y}} \le {\rm{y}}} \right\}\)in terms of unions and/or intersections of the three events \(\left\{ {{{\rm{X}}_{\rm{i}}} \le {\rm{y}}} \right\}\), \(\left\{ {{{\rm{X}}_{\rm{2}}} \le {\rm{y}}} \right\}\), and \(\left\{ {{{\rm{X}}_3} \le {\rm{y}}} \right\}\).)

b. Compute the expected system lifetime

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?
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