/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none}

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Q74E (page 243)

In an area having sandy soil,\({\rm{50}}\)small trees of a certain type were planted, and another \({\rm{50}}\) trees were planted in an area having clay soil. Let \({\rm{X = }}\) the number of trees planted in sandy soil that survive \({\rm{1}}\) year and \({\rm{Y = }}\)the number of trees planted in clay soil that survive \({\rm{1}}\) year. If the probability that a tree planted in sandy soil will survive \({\rm{1}}\)year is \({\rm{.7}}\)and the probability of \({\rm{1}}\)-year survival in clay soil is \({\rm{.6}}\), compute an approximation to \({\rm{P( - 5拢 X - Y拢 5)}}\) (do not bother with the continuity correction).

Short Answer

Expert verified

\({\rm{P( - 5拢 X - Y拢 5) = 0}}{\rm{.4826}}\)

Step by step solution

01

Definition of probability

the proportion of the total number of conceivable outcomes to the number of options in an exhaustive collection of equally likely outcomes that cause a given occurrence.

02

Determining the approximation to \({\rm{P( - 5£ X - Y£ 5)}}\)

Because an approximation of the probability is required, notice that random variables \({\rm{X}}\)and \({\rm{Y}}\)have a normal distribution. The mean value of the random variable \({\rm{X}}\)is

\({{\rm{\mu }}_{\rm{X}}}{\rm{ = 50 \times 0}}{\rm{.7 = 35}}\)

where \({\rm{50 }}\)is the number of little trees and the likelihood of a tree planted in sandy soil surviving a year is \({\rm{0}}{\rm{.7}}\), it is estimated that\({\rm{35}}\)total trees will survive (the expected value for a Binomial distribution is \({\rm{n \times p}}\)). The random variable \({\rm{Y}}\)has the same mean value.

\({{\rm{\mu }}_{\rm{Y}}}{\rm{ = 50 \times 0}}{\rm{.6 = 30}}\)

where \({\rm{50 }}\) is the number of little trees and the probability of a tree planted in clay soil surviving a year is \({\rm{0}}{\rm{.6 }}\), a total of\({\rm{30}}\)trees should survive.

The random variable \({\rm{X}}\)has a variance of

\({\rm{\sigma }}_{\rm{X}}^{\rm{2}}{\rm{ = 50 \times 0}}{\rm{.7 \times (1 - 0}}{\rm{.7) = 10}}{\rm{.5}}\)

\({\rm{n \times p \times (1 - p)}}\)is used to calculate the variance (as for Binomial distribution). The variance of the random variable \({\rm{Y}}\)is similar.

\({\rm{\sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ = 50 \times 0}}{\rm{.6 \times (1 - 0}}{\rm{.6) = 12}}\)

The random variable \({\rm{X - Y}}\)has an expected value of

\({{\rm{\mu }}_{{\rm{X - Y}}}}{\rm{ = E(X - Y) = E(X) - E(Y) = 35 - 30 = 5}}\)

The random variable \({\rm{X - Y}}\)has a variance of

\({\rm{\sigma }}_{{\rm{X - Y}}}^{\rm{2}}{\rm{ = }}{{\rm{1}}^{\rm{2}}}{\rm{\sigma }}_{\rm{X}}^{\rm{2}}{\rm{ + ( - 1}}{{\rm{)}}^{\rm{2}}}{\rm{\sigma }}_{\rm{Y}}^{\rm{2}}{\rm{ = 10}}{\rm{.5 + 12 = 22}}{\rm{.5}}\)

Finally, using a standard normal distribution, the requested probability can be estimated as follows:

\(\begin{array}{*{20}{c}}{{\rm{P( - 5拢 X - Y拢 5)}}}&{{\rm{ = P}}\left( {\frac{{{\rm{ - 5 - 5}}}}{{\sqrt {{\rm{22}}{\rm{.5}}} }}{\rm{拢 }}\frac{{{\rm{X - Y - }}{{\rm{\mu }}_{{\rm{X - Y}}}}}}{{{{\rm{\sigma }}_{{\rm{X - Y}}}}}}{\rm{拢 }}\frac{{{\rm{5 - 5}}}}{{\sqrt {{\rm{22}}{\rm{.5}}} }}} \right)}\\{}&{{\rm{ = P( - 2}}{\rm{.11拢 Z拢 0) = P(Z拢 0) - P(Z拢 - 2}}{\rm{.11)}}}\\{}&{\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{0}}{\rm{.5 - 0}}{\rm{.0174}}}\\{}&{{\rm{ = 0}}{\rm{.4826}}}\end{array}\)

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

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Refer to Exercise \({\rm{46}}\). Suppose the distribution is normal (the cited article makes that assumption and even includes the corresponding normal density curve).

a. Calculate \({\rm{P}}\)(\(69\text{拢}\bar{X}\text{拢}71\)) when \({\rm{n = 16}}\).

b. How likely is it that the sample mean diameter exceeds \({\rm{71}}\) when \({\rm{n = 25}}\)?

Suppose the amount of liquid dispensed by a certain machine is uniformly distributed with lower limit \({\rm{A = 8oz}}\) and upper limit\({\rm{B = 10oz}}\). Describe how you would carry out simulation experiments to compare the sampling distribution of the (sample) fourth spread for sample sizes\({\rm{n = 5,10,20}}\), and\({\rm{30}}\).

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. Let \({{\rm{X}}_{\rm{1}}}\)have a chi-squared distribution with parameter \({{\rm{\nu }}_{\rm{1}}}\) (see Section 4.4), and let \({{\rm{X}}_{\rm{2}}}\)be independent of \({{\rm{X}}_{\rm{1}}}\)and have a chi-squared distribution with parameter\({{\rm{v}}_{\rm{2}}}\). Use the technique of to show that \({{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}\)has a chi-squared distribution with parameter\({{\rm{v}}_{\rm{1}}}{\rm{ + }}{{\rm{v}}_{\rm{2}}}\).

b. You were asked to show that if \({\rm{Z}}\)is a standard normal \({\rm{rv}}\), then \({{\rm{Z}}^{\rm{2}}}\)has a chi squared distribution with\({\rm{v = 1}}\). Let \({{\rm{Z}}_{\rm{1}}}{\rm{,}}{{\rm{Z}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{Z}}_{\rm{n}}}\)be \({\rm{n}}\)independent standard normal \({\rm{rv}}\) 's. What is the distribution of\({\rm{Z}}_{\rm{1}}^{\rm{2}}{\rm{ + }}...{\rm{ + Z}}_{\rm{n}}^{\rm{2}}\)? Justify your answer.

c. Let \({{\rm{X}}_{\rm{1}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}\)be a random sample from a normal distribution with mean \({\rm{\mu }}\)and variance\({{\rm{\sigma }}^{\rm{2}}}\). What is the distribution of the sum \({\rm{Y = }}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{{\left( {\left( {{{\rm{X}}_{\rm{i}}}{\rm{ - \mu }}} \right){\rm{/\sigma }}} \right)}^{\rm{2}}}} {\rm{?}}\)Justify your answer?

Let \({\rm{A}}\)denote the percentage of one constituent in a randomly selected rock specimen, and let \({\rm{B}}\)denote the percentage of a second constituent in that same specimen. Suppose \({\rm{D}}\)and \({\rm{E}}\)are measurement errors in determining the values of \({\rm{A}}\)and \({\rm{B}}\)so that measured values are \({\rm{X = A + D}}\)and\({\rm{Y = B + E}}\), respectively. Assume that measurement errors are independent of one another and of actual values.

a. Show that

\({\rm{Corr(X,Y) = Corr(A,B) \times }}\sqrt {{\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right)} {\rm{ \times }}\sqrt {{\rm{Corr}}\left( {{{\rm{Y}}_{\rm{1}}}{\rm{,}}{{\rm{Y}}_{\rm{2}}}} \right)} \)

where \({{\rm{X}}_{\rm{1}}}\)and \({{\rm{X}}_{\rm{2}}}\)are replicate measurements on the value of\({\rm{A}}\), and \({{\rm{Y}}_{\rm{1}}}\)and \({{\rm{Y}}_{\rm{2}}}\)are defined analogously with respect to\({\rm{B}}\). What effect does the presence of measurement error have on the correlation?

b. What is the maximum value of \({\rm{Corr(X,Y)}}\)when \({\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.8100}}\)and \({\rm{Corr}}\left( {{{\rm{Y}}_{\rm{1}}}{\rm{,}}{{\rm{Y}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.9025?}}\)Is this disturbing?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.