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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

Two components of a minicomputer have the following joint pdf for their useful lifetimes \({\rm{X}}\)and \({\rm{Y}}\)

a. What is the probability that the lifetime \({\rm{X}}\) of the first component exceeds \({\rm{3}}\)?

b. What are the marginal pdf鈥檚 of \({\rm{X}}\)and \({\rm{Y}}\)? Are the two lifetimes independent? Explain.

c. What is the probability that the lifetime of at least one component exceeds\({\rm{3}}\)?

In cost estimation, the total cost of a project is the sum of component task costs. Each of these costs is a random variable with a probability distribution. It is customary to obtain information about the total cost distribution by adding together characteristics of the individual component cost distributions鈥攖his is called the 鈥渞oll-up鈥� procedure. For example, \(E\left( {{X_1} + .. + {X_n}} \right) = E\left( {{X_1}} \right) + .. + E\left( {{X_n}} \right)\), so the roll-up procedure is valid for mean cost. Suppose that there are two component tasks that \({X_1} and {X_2}\)are independent, normally distributed random variables. Is the roll-up procedure valid for the \(7{5^{th}}\) percentile? That is, is the \(7{5^{th}}\) percentile of the distribution of\({X_1} + {X_2}\)the same as the sum of the \(7{5^{th}}\) percentiles of the two individual distributions? If not, what is the relationship between the percentile of the sum and the sum of percentiles? For what percentiles is the roll-up procedure valid in this case?

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 \({\rm{\rho ?}}\)

There are two traffic lights on a commuter's route to and from work. Let \({{\rm{X}}_{\rm{1}}}\) be the number of lights at which the commuter must stop on his way to work, and \({{\rm{X}}_{\rm{2}}}\) be the number of lights at which he must stop when returning from work. Suppose these two variables are independent, each with pmf given in the accompanying table (so \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\) is a random sample of size \({\rm{n = 2}}\)).

a. Determine the pmf of \({{\rm{T}}_{\rm{o}}}{\rm{ = }}{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}\).

b. Calculate \({{\rm{\mu }}_{{{\rm{T}}_{\rm{o}}}}}\). How does it relate to \({\rm{\mu }}\), the population mean?

c. Calculate \({\rm{\sigma }}_{{{\rm{T}}_{\rm{o}}}}^{\rm{2}}\). How does it relate to \({{\rm{\sigma }}^{\rm{2}}}\), the population variance?

d. Let \({{\rm{X}}_{\rm{3}}}\) and \({{\rm{X}}_{\rm{4}}}\) be the number of lights at which a stop is required when driving to and from work on a second day assumed independent of the first day. With \({{\rm{T}}_{\rm{o}}}{\rm{ = }}\) the sum of all four \({{\rm{X}}_{\rm{i}}}\) 's, what now are the values of \({\rm{E}}\left( {{{\rm{T}}_{\rm{a}}}} \right)\) and \({\rm{V}}\left( {{{\rm{T}}_{\rm{a}}}} \right)\)?

e. Referring back to (d), what are the values of \({\rm{P}}\left( {{{\rm{T}}_{\rm{o}}}{\rm{ = 8}}} \right)\) and \(\text{P}\left( {{\text{T}}_{\text{e}}}\text{ }\!\!{}^\text{3}\!\!\text{ 7} \right)\) (Hint: Don't even think of listing all possible outcomes!)

a. Use the general formula for the variance of a linear combination to write an expression for\({\rm{V(aX + Y)}}\). Then let\({\rm{a = }}{{\rm{\sigma }}_{\rm{\gamma }}}{\rm{/}}{{\rm{\sigma }}_{\rm{X}}}\), and show that. (Hint: Variance is always, and\({\rm{Cov(X,Y) = }}{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{Y}}}{\rm{ \times \rho }}\).)

b. By considering\({\rm{V(aX - Y)}}\), conclude that\({\rm{\rho 拢 1}}\).

c. Use the fact that \({\rm{V(W) = 0}}\)only if \({\rm{W}}\)is a constant to show that \({\rm{\rho = 1}}\)only if\({\rm{Y = aX + b}}\).
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