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Chapter 5: Q60E (page 241)

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\(P(Y > - 2)\).

Short Answer

Expert verified

\(\begin{array}{l}P(Y \ge 0) = 0.2877\\P( - 1 \le Y \le 1) = 0.3686\end{array}\)

Step by step solution

01

Definition of Standard Deviation

The standard deviation is a statistical measure of how much a set of values varies or disperses. A low standard deviation implies that the values are close to the set's mean (also known as the anticipated value), whereas a large standard deviation shows that the values are spread out over a wider range.

02

Calculation for finding the value of probability.

The given random variable can be written as

\(Y = \frac{1}{2}{X_1} + \frac{1}{2}{X_2} - \frac{1}{3}{X_3} - \frac{1}{3}{X_4} - \frac{1}{3}{X_5}\)

It is normally distributed with mean value of

\(\begin{aligned}E(Y) &= E\left( {\frac{1}{2}{X_1} + \frac{1}{2}{X_2} - \frac{1}{3}{X_3} - \frac{1}{3}{X_4} - \frac{1}{3}{X_5}} \right)\\ &= \frac{1}{2}E\left( {{X_1}} \right) + \frac{1}{2}E\left( {{X_2}} \right) - \frac{1}{3}E\left( {{X_3}} \right) - \frac{1}{3}E\left( {{X_4}} \right) - \frac{1}{3}E\left( {{X_5}} \right)\\ &= - 1\end{aligned}\)

(1): the expected values are given in the mentioned example.

The variance of random variable Y is

\(\begin{aligned}V(Y) &= V\left( {\frac{1}{2}{X_1} + \frac{1}{2}{X_2} - \frac{1}{3}{X_3} - \frac{1}{3}{X_4} - \frac{1}{3}{X_5}} \right)\\ &= \frac{1}{4}V\left( {{X_1}} \right) + \frac{1}{4}V\left( {{X_2}} \right) + \frac{1}{9}V\left( {{X_3}} \right) + \frac{1}{9}V\left( {{X_4}} \right) + \frac{1}{9}V\left( {{X_5}} \right)\\ &= 3.167\end{aligned}\)

(2): the variances are given in the mentioned example.

03

Calculation for finding the value of probability.

The standard deviation of random variable Y is

\({\sigma _Y} = \sqrt {V(X)} = \sqrt {3.167} = 1.7795\)

The probability of event \(\{ Y \ge 0\} \)is

\(\begin{aligned}P(Y \ge 0) &= P\left( {\frac{{Y - E(Y)}}{{{\sigma _Y}}} \ge \frac{{0 - ( - 1)}}{{1.7795}}} \right)\\ &= P(Z \ge 0.56)\\ &= 1 - P(Z < 0.56)\\ &= 0..2877\end{aligned}\)

(3): from the normal probability table in the appendix. The probability can also be computed with software.

04

Calculation for finding the value of probability.

Similarly, the probability of the event \(\{ - 1 \le Y \le 1\} \)is

\(\begin{aligned}P( - 1 \le Y \le 1) &= P\left( {\frac{{ - 1 - ( - 1)}}{{1.7795}} \le Z \le \frac{{1 - ( - 1)}}{{1.7795}}} \right)\\ &= P(0 \le Z \le 1.12)\\ &= 0.3686,\end{aligned}\)

(3): from the normal probability table in the appendix. The probability can also be computed with a software.

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Most popular questions from this chapter

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\({\rm{\mu }}\)?

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

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?

Consider a small ferry that can accommodate cars and buses. The toll for cars is\({\rm{\$ 3}}\), and the toll for buses is\({\rm{\$ 10}}\). Let \({\rm{X}}\) and \({\rm{Y}}\) denote the number of cars and buses, respectively, carried on a single trip. Suppose the joint distribution of \({\rm{X}}\) and\({\rm{Y}}\). Compute the expected revenue from a single trip.

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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