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CHAPTER 4 Random Variables and Probability Distributions

Generating the Sampling Distribution of x

Generating the Sampling Distribution of x

Select sample size n (large) from target population

Population: Mean — // Std. Dev. = a Unknown shape

Repeat this

process an

infinite

number

of times

Sampling distribution of x

(i.e., theoretical population of x's)

Mean = /j,^ = p,

Std. Dev. = ctj = oA/n

Normal distribution (Central Limit Theorem)

Supplementary Exercises 4.170-4.204

Note: Starred (*) exercises refer to the optional section in this chapter.

Understanding the Principles

4.170 Which of the following describe discrete random variables, and which describe continuous random variables?

a. The length of time that an exercise physiologist's pro gram takes to elevate her client's heart rate to 140 beats per minute

b. The number of crimes committed on a college campus per year

c. The number of square feet of vacant office space in a large city

d. The number of voters who favor a new tax proposal

4.171 For each of the following examples, decide whether x is a binomial random variable and explain your decision:

a. A manufacturer of computer chips randomly selects 100 chips from each hour's production in order to estimate the proportion of defectives. Let x represent the num ber of defectives in the 100 chips sampled.

b. Of five applicants for a job, two will be selected. Although all applicants appear to be equally qualified, only three have the ability to fulfill the expectations of the company. Suppose that the two selections are made at random from the five applicants, and let x be the number of qualified applicants selected.

c. A software developer establishes a support hot line for customers to call in with questions regarding use of the software. Let x represent the number of calls received on the hot line during a specified workday.

d. Florida is one of a minority of states with no state income tax. A poll of 1,000 registered voters is conducted to determine how many would favor a state income tax in light of the state's current fiscal condition. Let x be the number in the sample who would favor the tax.

4.172 Describe how you could obtain the simulated sampling distribution of a sample statistic.

4.173 True or False. The sample mean, x, will always be equal

4.174 True or False. The sampling distribution of x is normally distributed, regardless of the size of the sample n.

Learning the Mechanics

4.175 Suppose x is a binomial random variable with n = 20 and | p= .7.

a. FindP(x = 14).

b. FindP(x £ 12).

c. Find P(x > 12).

d. FindP(9 < x < 18).

e. FindP(8 < x < 18).

f. Find n, cr , and a.

g. What is the probability that x is in the interval | ju ± 2cr?

4.176 Consider the discrete probability distribution shown here: I

a. Calculate p,, cr2, and u.

b. WhatisP(x < 15)?

c. Calculate ju ± 2<r.

d. What is the probability that x is in the interval /n ± 2<r?

4.177 The random variable x has a normal distribution will /x = 70 and a = 10. Find the following probabilities:

a. P(x < 75)

b. P(x ^ 90)

c. P(60 < x < 75)

d. P(x > 75)

e. P(x = 75)

f. P(x < 95)

4.178 The random variable x has a normal distribution wilt I /x = 40 and a2 = 36. Find a value of x, say, x0, such that |

a. P(x > x0) = .5

b. P(x s x0) = .9911

c. P(x < x0) = .0028

d. P(x > x0) = .0228

e. P(x < x0) = .1003

f. P(x > x()) = .7995

*4.179 Assume that x is a binomial random variable with n = ll and p = .5. Use the normal probability distribution 111 approximate the following probabilities:

a. P(x < 48)

b. P(50 < x < 65)