/*! 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} Q106SE A plan for an executive travelle... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Q106SE (page 138)

A plan for an executive travellers鈥� club has been developed by an airline on the premise that \({\rm{10\% }}\) of its current customers would qualify for membership.

a. Assuming the validity of this premise, among \({\rm{25}}\) randomly selected current customers, what is the probability that between \({\rm{2}}\) and \({\rm{6}}\) (inclusive) qualify for membership?

b. Again assuming the validity of the premise, what are the expected number of customers who qualify and the standard deviation of the number who qualify in a random sample of \({\rm{100}}\) current customers?

c. Let \({\rm{X}}\) denote the number in a random sample of \({\rm{25}}\) current customers who qualify for membership. Consider rejecting the company鈥檚 premise in favour of the claim that \({\rm{p > 10}}\) if \({\rm{x}} \ge {\rm{7}}\). What is the probability that the company鈥檚 premise is rejected when it is actually valid?

d. Refer to the decision rule introduced in part (c). What is the probability that the company鈥檚 premise is not rejected even though \({\rm{p = }}{\rm{.20}}\) (i.e., \({\rm{20\% }}\) qualify)?

Short Answer

Expert verified

(a) The probability that between\({\rm{2}}\)and\(6\)(inclusive) customers who will qualify for membership is\({\rm{0}}{\rm{.720}}\).

(b) The expected number of customers who qualify and the standard deviation of the number who qualify in a random sample of\(100\)current customers is\({\rm{10}}\).

(c) The probability that the company鈥檚 premise is rejected when it is actually valid is\({\rm{0}}{\rm{.009}}\).

(d) The probability that the company鈥檚 premise is not rejected even though \({\rm{p = }}{\rm{.20}}\) qualify is \({\rm{0}}{\rm{.78}}\).

Step by step solution

01

Concept Introduction

Probability refers to the likelihood of a random event's outcome. This word refers to determining the likelihood of a given occurrence occurring.

The Binomial Random Variable\({\rm{X}}\)is defined as

\({\rm{X = }}\)the number of customers who qualify for membership out of\({\rm{n}}\)

Where conditions one to four on page\({\rm{117}}\)are satisfied (binomial experiment).

The described random variable has Binomial Distribution with parameters

\({\rm{n}}\)and\({\rm{p}}\)where\({\rm{p = 0}}{\rm{.1}}\).

02

Probability between \({\rm{2}}\) and \({\rm{6}}\)

(a)

There are\({\rm{25}}\)randomly selected current customers, which means that 鈥�

\({\rm{X}} \sim {\rm{Bin(25,0}}{\rm{.1)}}\)

Cumulative Density Function cdf of binomial random variable\({\rm{X}}\)with parameters\({\rm{n}}\)and\({\rm{p}}\)is 鈥�

\({\rm{B(x;n,p) = P(X}} \le {\rm{x) = }}\sum\limits_{{\rm{y = 0}}}^{\rm{x}} {\rm{b}} {\rm{(y;n,p),}}\;\;\;{\rm{x = 0,1, \ldots ,n}}\)

The theorem is 鈥�

\({\rm{b(x;n,p) = }}\left\{ {\begin{array}{*{20}{l}}{\left( {\begin{array}{*{20}{l}}{\rm{n}}\\{\rm{x}}\end{array}} \right){{\rm{p}}^{\rm{x}}}{{{\rm{(1 - p)}}}^{{\rm{n - x}}}}}&{{\rm{,x = 0,1,2, \ldots ,n}}}\\{\rm{0}}&{{\rm{, otherwise }}}\end{array}} \right.\)

The following is true 鈥�

\(\begin{aligned}{\rm{P(2}} \le {\rm{X}} \le 6) &= B(6;25,0) {\rm{.1 - B(1;25,0)}}{\rm{.1}}\\ &= 0 {\rm{.991 - 0}}{\rm{.271}}\\ &= 0 {\rm{.720}}\end{aligned}\)

Therefore, the value is obtained as \({\rm{0}}{\rm{.720}}\).

03

Number of customers to qualify

(b)

Proposition: For a binomial random variable \({\rm{X}}\) with parameters \(n,p,\) and \({\rm{q = 1 - p}}\), the following is true 鈥�

\(\begin{aligned}E(X) &= np; \\ V(X) &= np(1 - p) = npq;\\{{\rm{\sigma }}_{\rm{X}}} &= \sqrt {{\rm{npq}}} \end{aligned}\)

Since in this it is given 鈥�

\({\rm{X\sim Bin(100,0}}{\rm{.1)}}\)

The following holds 鈥�

\(\begin{aligned} E(X) = np &= 100 \cdot {\rm{0}}{\rm{.1 = 10}}\\ V(X) &= npq = 100 \cdot {\rm{0}}{\rm{.1}} \cdot {\rm{0}}{\rm{.9 = 9}}\\{{\rm{\sigma }}_{\rm{X}}} &= \sqrt {{\rm{V(X)}}} {\rm{ = }}\sqrt {\rm{9}} {\rm{ = 3}}\end{aligned}\)

Therefore, the value is obtained as \({\rm{E(X) = 10}}\).

04

Probability of company’s premise being rejected

(c)

Find the following probability 鈥�

\(\begin{gathered}P(X \geqslant 7{\text{ }}when{\text{ }}p = 0.1) &= 1 - B(6;25,0.1) \\ &= 1 - 0.991 = 0.009 \\ \end{gathered} \)

Therefore, the value is obtained as \({\rm{0}}{\rm{.009}}\).

05

Probability of company’s premise being rejected

(d)

Do not reject the premise when 鈥�

\({\rm{x}} \le {\rm{6}}\)

Which is complement of 鈥�

\({\rm{x}} \ge {\rm{7}}\)

Thus, find the probability of event 鈥�

\({\rm{\{ X}} \le {\rm{6 when p = 0}}{\rm{.2\} }}\)

Which is 鈥�

\({\rm{P(X}} \le {\rm{6 when p = 0}}{\rm{.2) = B(6;25,0}}{\rm{.2) = 0}}{\rm{.78}}\)

Therefore, the value is obtained as \({\rm{0}}{\rm{.78}}\).

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

Some parts of California are particularly earthquake prone. Suppose that in one metropolitan area, 25% of all homeowners are insured against earthquake damage. Four homeowners are to be selected at random; let X

denote the number among the four who have earthquake insurance.

a. Find the probability distribution of X. (Hint: Let S denote a homeowner who has insurance and F one who does not. Then one possible outcome is SFSS,

with probability (.25)(.75)(.25)(.25) and associated X value 3. There are 15 other outcomes.)

b. Draw the corresponding probability histogram.

c. What is the most likely value for X?

d. What is the probability that at least two of the four selected have earthquake insurance?

Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of\({\bf{10}}\)organisms/m3 (the article 鈥淐ounting at Low Concentrations: The Statistical Challenges of Verifying Ballast Water Discharge Standards鈥� (Ecological Applications) considers using the Poisson process for this purpose). a. What is the probability that one cubic meter of discharge contains at least 8 organisms? b. What is the probability that the number of organisms in\({\bf{1}}.{\bf{5}}{\rm{ }}{{\bf{m}}^3}\)of discharge exceeds its mean value by more than one standard deviation? c. For what amount of discharge would the probability of containing at least one organism be\(.{\bf{999}}\)?

Twenty pairs of individuals playing in a bridge tournament have been seeded \({\rm{1,}}....{\rm{,20}}\). In the first part of the tournament, the \({\rm{20}}\) are randomly divided into 10 east鈥� west pairs and \({\rm{10}}\) north鈥搒outh pairs. a. What is the probability that x of the top \({\rm{10}}\) pairs end up playing east鈥搘est? b. What is the probability that all of the top five pairs end up playing the same direction? c. If there are \({\rm{2}}\)n pairs, what is the pmf of X = the number among the top n pairs who end up playing east鈥搘est? What are E(X) and V(X)?

A second-stage smog alert has been called in a certain area of Los Angeles County in which there are \({\rm{50}}\) industrial firms. An inspector will visit \({\rm{10}}\) randomly selected firms to check for violations of regulations. a. If \({\rm{15}}\) of the firms are actually violating at least one regulation, what is the pmf of the number of firms visited by the inspector that are in violation of at least one regulation? b. If there are \({\rm{500}}\) firms in the area, of which \({\rm{150}}\) are in violation, approximate the pmf of part (a) by a simpler pmf. c. For X = the number among the 10 visited that are in violation, compute E(X) and V(X) both for the exact pmf and the approximating pmf in part (b).

a. Show that b(x; n,\({\rm{1 - }}\)p) = b(n - x; n, p). b. Show that B(x; n,\({\rm{1 - }}\)p) =\({\rm{1 - }}\)B(n - x -\({\rm{1}}\); n, p). (Hint: At most x S鈥檚 is equivalent to at least (n - x) F鈥檚.) c. What do parts (a) and (b) imply about the necessity of including values of p greater than\({\rm{.5}}\)in Appendix Table A\({\rm{.1}}\)?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

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.