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Chapter 5: Q 5.110 (page 224)

A variable y of a finite population has the following frequency distribution:

y1234
f22106

Suppose a member is selected at random from the population and let Y denote the value of the variable y for the member obtained.

a. Determine the probability distribution of the random variable Y.

b. Use random-variable notation to describe the events that Y takes on the value 3, a value less than 3, and a value of at least 3.

c. Find P(Y = 3), P(Y <3), and P(Y 鈮� 3). Interpret your results.

d. Construct a probability histogram for the random variable Y.

Short Answer

Expert verified

Part a.

Random Variable (Y)

Probability

1

2/20 = 0.1

2

2/20 = 0.1

3

10/20 = 0.5

4

6/20 = 0.3

Part b.

On the value of 3: Y = 3

A value less than 3: Y < 3

A value of at least 3 : Y鈮� 3

Part c.

P(Y=3) = 0.5

P(Y < 3) = 0.1 + 0.1 = 0.2

P(Y 鈮� 3) = 0.5 + 0.3 = 0.8

Part d.

Step by step solution

01

Part (a) Step 1. Given information

The frequency distribution of a variable x in a finite population is as follows: :

y

1

2

3

4

f

2

2

10

6

Assume a member is chosen at random from the population, and Y is the value of the variable y for that member.

02

Part (a) Step 2. Formula Used 

The experiment is conducted out assuming that the random variable (X=x) occurs n times out of a total of N times. As a result of the f/N rule:

P (X=x) = n/N

03

Part (a) Step 3. Solution

Here, N = 2+2+10+6 = 20. As a result, the probability of random variable Y is as follows:

Random Variable

Probability

1

2/20 = 0.1

2

2/20 = 0.1

3

10/20 = 0.5

4

6/20 = 0.3

04

Part (b) Step 1. Solution 

Event where Y takes the value 3, can also be written as:

Y = 3

Event where Y takes a value less than 3, can also be written as :

Y < 3

Event where Y takes a value of at least 3, can also be written as :

Y 鈮� 3

05

Part (c) Step 1. Solution

From the table,

P(Y = 3) = 0.5

P(Y < 3) = 0.1 + 0.1 = 0.2

P(Y 鈮� 3 ) = 0.5 + 0.3 = 0.8

06

Part (d) Step 1. Solution

The probability histogram for the random variable Y is as follows:

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Most popular questions from this chapter

Jurors. From 10 men and 8 women in a pool of potential jurors 12 are chosen at random to constitute a jury. Suppose that you observe the number of men who are chosen for the jury. Let A be the event that at least half of the 12 jurors are men, and let B be the event that at least half of the 8 women are on the jury.

Part (a) Determine the sample space for this experiment

Part (b) Find (A or B) ,(A & B) and (A &(not B)), listing all the outcomes for each of those three events

Part (c) Are events A and B are mutually exclusive.? are events A and (not B)? are events (not A) and (not B)? Explain.

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(a) A is a quantitative variable whose value depends on chance.

(b) A discrete random variable is a random variable whose possible values .

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What is the difference between selecting a member at random from a finite population and taking a simple random sample of size 1?

Suppose that a simple random sample is taken from a finite population in which each member is classified as either having or not having a specified attribute. Fill in the following blanks.

(a) If sampling is with replacement, the probability distribution of the number of members sampled that have the specified attribute is a distribution.

(b) If sampling is without replacement, the probability distribution of the number of members sampled that have the specified attribute is a distribution.

(c) If sampling is without replacement and the sample size does not exceed % of the population size, the probability distribution of the number of members sampled that have the specified attribute can be approximated by a distribution.
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