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Chapter 5: Q72E (page 243)

I have three errands to take care of in the Administration Building. Let \({\rm{Xi = }}\)the time that it takes for the \({\rm{ ith}}\)errand\({\rm{(i = 1,2,3)}}\), and let \({{\rm{X}}_{\rm{4}}}{\rm{ = }}\)the total time in minutes that I spend walking to and from the building and between each errand. Suppose the \({\rm{Xi`s}}\)are independent, and normally distributed, with the following means and standard deviations: \({{\rm{\mu }}_{{{\rm{X}}_{\rm{1}}}}}{\rm{ = 15,}}{{\rm{\sigma }}_{{{\rm{X}}_{\rm{1}}}}}{\rm{ = 4,}}{{\rm{\mu }}_{{{\rm{X}}_{\rm{2}}}}}{\rm{ = 5,}}{{\rm{\sigma }}_{{{\rm{X}}_{\rm{2}}}}}{\rm{ = 1,}}{{\rm{\mu }}_{{{\rm{X}}_{\rm{3}}}}}{\rm{ = 8,}}{{\rm{\sigma }}_{{{\rm{X}}_{\rm{3}}}}}{\rm{ = 2,}}{{\rm{\mu }}_{{{\rm{X}}_{\rm{4}}}}}{\rm{ = 12,}}{{\rm{\sigma }}_{{{\rm{X}}_{\rm{4}}}}}{\rm{ = 3}}\).plan to leave my office at precisely \({\rm{10:00}}\)a.m. and wish to post a note on my door that reads, 鈥淚 will return by \({\rm{t}}\)a.m.鈥� What time\({\rm{t}}\)should I write down if I want the probability of my arriving after \({\rm{t}}\) to be \({\rm{.01}}\)?

Short Answer

Expert verified

\({\rm{10:53\;A}}{\rm{.M}}{\rm{.\;}}\)

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

Given:

\(\begin{array}{*{20}{c}}{{{\rm{\mu }}_{{{\rm{X}}_{\rm{1}}}}}{\rm{ = 15}}}\\{{{\rm{\mu }}_{{{\rm{X}}_{\rm{2}}}}}{\rm{ = 5}}}\\{{{\rm{\mu }}_{{{\rm{X}}_{\rm{3}}}}}{\rm{ = 8}}}\\{{{\rm{\mu }}_{{{\rm{X}}_{\rm{4}}}}}{\rm{ = 12}}}\\{{{\rm{\sigma }}_{{{\rm{X}}_{\rm{1}}}}}{\rm{ = 4}}}\\{{{\rm{\sigma }}_{{{\rm{X}}_{\rm{2}}}}}{\rm{ = 1}}}\\{{{\rm{\sigma }}_{{{\rm{X}}_{\rm{3}}}}}{\rm{ = 2}}}\\{{{\rm{\sigma }}_{{{\rm{X}}_{\rm{4}}}}}{\rm{ = 3}}}\end{array}\)

The mean, variance, and standard deviation for the linear combination \({\rm{W = a}}{{\rm{X}}_{\rm{1}}}{\rm{ + b}}{{\rm{X}}_{\rm{2}}}\)are as follows:

\(\begin{array}{*{20}{c}}{{{\rm{\mu }}_{\rm{W}}}{\rm{ = a}}{{\rm{\mu }}_{\rm{1}}}{\rm{ + b}}{{\rm{\mu }}_{\rm{2}}}}\\{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}{\rm{ = }}{{\rm{a}}^{\rm{2}}}{\rm{\sigma }}_{\rm{1}}^{\rm{2}}{\rm{ + }}{{\rm{b}}^{\rm{2}}}{\rm{\sigma }}_{\rm{2}}^{\rm{2}}\left( {{\rm{\;If\;}}{{\rm{X}}_{\rm{ - }}}{\rm{1\;and\;}}{{\rm{X}}_{\rm{ - }}}{\rm{2\;are independent\;}}} \right)}\\{{{\rm{\sigma }}_{\rm{W}}}{\rm{ = }}\sqrt {{{\rm{a}}^{\rm{2}}}{\rm{\sigma }}_{\rm{1}}^{\rm{2}}{\rm{ + }}{{\rm{b}}^{\rm{2}}}{\rm{\sigma }}_{\rm{2}}^{\rm{2}}} {\rm{\;(If\;}}{{\rm{X}}_{\rm{ - }}}{\rm{1\;and\;}}{{\rm{X}}_{\rm{ - }}}{\rm{2\;are independent)\;}}}\end{array}\)

The total time is the sum of all times \({{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}{\rm{ + }}{{\rm{X}}_{\rm{4}}}\) (including time spent going to and from the building and between each errand):

\(\begin{array}{*{20}{c}}{{{\rm{\mu }}_{{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}{\rm{ + }}{{\rm{X}}_{\rm{4}}}}}{\rm{ = }}{{\rm{\mu }}_{{{\rm{X}}_{\rm{1}}}}}{\rm{ + }}{{\rm{\mu }}_{{{\rm{X}}_{\rm{2}}}}}{\rm{ + }}{{\rm{\mu }}_{{{\rm{X}}_{\rm{3}}}}}{\rm{ + }}{{\rm{\mu }}_{{{\rm{X}}_{\rm{4}}}}}{\rm{ = 15 + 5 + 8 + 12 = 40}}}\\{{{\rm{\sigma }}_{{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}{\rm{ + }}{{\rm{X}}_{\rm{4}}}}}{\rm{ = }}\sqrt {{\rm{\sigma }}_{{{\rm{X}}_{\rm{1}}}}^{\rm{2}}{\rm{ + \sigma }}_{{{\rm{X}}_{\rm{2}}}}^{\rm{2}}{\rm{ + \sigma }}_{{{\rm{X}}_{\rm{3}}}}^{\rm{2}}{\rm{ + \sigma }}_{{{\rm{X}}_{\rm{4}}}}^{\rm{2}}} {\rm{ = }}\sqrt {{{{\rm{(4)}}}^{\rm{2}}}{\rm{ + (1}}{{\rm{)}}^{\rm{2}}}{\rm{ + (2}}{{\rm{)}}^{\rm{2}}}{\rm{ + (3}}{{\rm{)}}^{\rm{2}}}} {\rm{ = }}\sqrt {{\rm{30}}} {\rm{\gg 5}}{\rm{.4772}}}\end{array}\)

The cutoff value for the highest \({\rm{1\% }}\) all time is \({\rm{99\% }}\)of all time. Calculate the $z$-score associated with a probability of \({\rm{0}}{\rm{.99\% }}\)or \({\rm{99\% }}\):

\({\rm{z = 2}}{\rm{.33}}\)

The appropriate number is the mean multiplied by the standard deviation and the z-score:

\({\rm{x = \mu + z\sigma = 40 + 2}}{\rm{.33}}\sqrt {{\rm{30}}} {\rm{\gg 53 99\% }}\)

As a result, the likelihood of coming after \({\rm{53}}\) minutes is \({\rm{0}}{\rm{.01,53}}\)minutes after \({\rm{10:53\;A}}{\rm{.M}}{\rm{.\;}}\).

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Most popular questions from this chapter

A surveyor wishes to lay out a square region with each side having length\({\rm{L}}\). However, because of a measurement error, he instead lays out a rectangle in which the north-south sides both have length \({\rm{X}}\) and the east-west sides both have length\({\rm{Y}}\). Suppose that \({\rm{X}}\) and \({\rm{Y}}\) are independent and that each is uniformly distributed on the interval \({\rm{(L - A,L + A)}}\) (where \({\rm{0 < A < L}}\) ). What is the expected area of the resulting rectangle?

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).)

A box contains ten sealed envelopes numbered\({\rm{1, \ldots ,10}}\). The first five contain no money, the next three each contains\({\rm{\$ 5}}\), and there is a \({\rm{\$ 10}}\) bill in each of the last two. A sample of size \({\rm{3}}\) is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. If \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and \({{\rm{X}}_{\rm{3}}}\) denote the amounts in the selected envelopes, the statistic of interest is \({\rm{M = }}\) the maximum of\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and\({{\rm{X}}_{\rm{3}}}\).

a. Obtain the probability distribution of this statistic.

b. Describe how you would carry out a simulation experiment to compare the distributions of \({\rm{M}}\) for various sample sizes. How would you guess the distribution would change as \({\rm{n}}\) increases?

Refer back to Example, Two cars with six-cylinder engines and three with four-cylinder engines are to be driven over a \(300\)-mile course. Let \({X_1}, . . . {X_5}\)denote the resulting fuel efficiencies (mpg). Consider the linear combination

\(Y = \left( {{X_1} + {X_2}} \right)/2 - \left( {{X_3} + {X_4} + {X_5}} \right)/3\)

which is a measure of the difference between four-cylinder and six-cylinder vehicles. Compute \(P\left( {0 \le Y} \right)\)and\(P(Y > - 2)\).

Suppose that when the pH of a certain chemical compound is\(5.00\), the pH measured by a randomly selected beginning chemistry student is a random variable with a mean of\(5.00\)and a standard deviation .2. A large batch of the compound is subdivided and a sample is given to each student in a morning lab and each student in an afternoon lab. Let\(X = \)the average pH as determined by the morning students and\(Y = \)the average pH as determined by the afternoon students.

a. If pH is a normal variable and there are\(25\)students in each lab, compute\(P\left( { - .1 \le X - Y \le - .1} \right)\)

b. If there are\(36\)students in each lab, but pH determinations are not assumed normal, calculate (approximately)\(P\left( { - .1 \le X - Y \le - .1} \right)\).
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