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Running Head: DATA COLLECTION 1

DATA COLLECTION 2

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ANALYSIS OF VARIANCE 387

INTRODUCTION

In this chapter, we continue our discussion of hypothesis testing. Recall that in Chapters 10

and 11 we examined the general theory of hypothesis testing. We described the case

where a sample was selected from the population. We used the z distribution (the standard

normal distribution) or the t distribution to determine whether it was reasonable to

conclude that the population mean was equal to a specified value. We tested whether

two population means are the same. In this chapter, we expand our idea of hypothesis

tests. We describe a test for variances and then a test that simultaneously compares

several population means to determine if they are equal.

COMPARING TWO POPULATION VARIANCES

In Chapter 11, we tested hypotheses about equal population means. The tests differed

based on our assumptions regarding whether the population standard deviations or

variances were equal or unequal. In this chapter, the assumption about equal population

variances is also important. In this section, we present a way to statistically test this

The F Distribution

The probability distribution used in this chapter is the F distribution. It was named to honor

Sir Ronald Fisher, one of the founders of modern-day statistics. The test statistic for several

situations follows this probability distribution. It is used to test whether two samples are

from populations having equal variances, and it is also applied when we want to compare

several population means simultaneously. The simultaneous comparison of several population

means is called analysis of variance (ANOVA). In both of these situations, the populations

must follow a normal distribution, and the data must be at least interval-scale.

What are the characteristics of the F distribution?

1. There is a family of F distributions. A particular member of the family is determined

by two parameters: the degrees of freedom in the numerator and the degrees of

freedom in the denominator. The shape of the distribution is illustrated by the following

graph. There is one F distribution for the combination of 29 degrees of freedom

in the numerator (df ) and 28 degrees of freedom in the denominator. There is

another F distribution for 19 degrees of freedom in the numerator and 6 degrees of

freedom in the denominator. The final distribution shown has 6 degrees of freedom

in the numerator and 6 degrees of freedom in the denominator. We will describe

the concept of degrees of freedom later in the chapter. Note that the shapes of the

distributions change as the degrees of freedom change.

The F distribution is continuous. This means that the value of F can assume an

infinite number of values between zero and positive infinity.

3. The F statistic cannot be negative. The smallest value F can assume is 0.

4. The F distribution is positively skewed. The long tail of the distribution is to the

right-hand side. As the number of degrees of freedom increases in both the numerator

and denominator, the distribution approaches a normal distribution.

5. The F distribution is asymptotic. As the values of F increase, the distribution approaches

the horizontal axis but never touches it. This is similar to the behavior of

the normal probability distribution, described in Chapter 7.

Testing a Hypothesis of Equal Population Variances

The first application of the F distribution that we describe occurs when we test the hypothesis

that the variance of one normal population equals the variance of another

normal population. The following examples will show the use of the test:

• A health services corporation manages two hospitals in Knoxville, Tennessee:

St. Mary’s North and St. Mary’s South. In each hospital, the mean waiting time in the

Emergency Department is 42 minutes. The hospital administrator believes that the

St. Mary’s North Emergency Department has more variation in waiting time than

St. Mary’s South.

• The mean rate of return on two types of

common stock may be the same, but

there may be more variation in the rate

of return in one than the other. A sample

of 10 technology and 10 utility

stocks shows the same mean rate of

return, but there is likely more variation

in the technology stocks.

• An on-line newspaper found that men

and women spend about the same

amount of time per day accessing

news apps. However, the same report

indicated the times of men had nearly

twice as much variation compared to

the times of women.

The F distribution is also used to test the assumption that the variances of two normal

populations are equal. Recall that in the previous chapter the t test to investigate

whether the means of two independent populations differed assumes that the variances

of the two normal populations are the same. See this list of assumptions on page 361.

The F distribution is used to test the assumption that the variances are equal.

To compare two population variances, we first state the null hypothesis. The null

hypothesis is that the variance of one normal population, σ21

, equals the variance of another

normal population, σ22

. The alternate hypothesis is that the variances differ. In this

instance, the null hypothesis and the alternate hypothesis are:

H0: σ2

1 = σ2

2

H1: σ2

1 ≠ σ2

2

To conduct the test, we select a random sample of observations, n1, from one population

and a random sample of observations, n2, from the second population. The test

statistic is defined as follows.

ANALYSIS OF VARIANCE 389

The terms s1

2 and s2

2 are the respective sample variances. If the null hypothesis is

true, the test statistic follows the F distribution with n1 − 1 and n2 − 1 degrees of freedom.

To reduce the size of the table of critical values, the larger sample variance is

placed in the numerator; hence, the tabled F ratio is always larger than 1.00. Thus, the

right-tail critical value is the only one required. The critical value of F for a two-tailed test

is found by dividing the significance level in half (α/2) and then referring to the appropriate

degrees of freedom in Appendix B.6. An example will illustrate.

SOLUTION

The mean driving times along the two routes are nearly the same. The mean time is

58.29 minutes for the U.S. 25 route and 59.0 minutes along the I-75 route. However,

in evaluating travel times, Mr. Lammers is also concerned about the variation

in the travel times. The first step is to compute the two sample variances. We’ll use

formula (3–9) to compute the sample standard deviations. To obtain the sample

variances, we square the standard deviations.

U.S. ROUTE 25

x =

Σx

n =

408

7 = 58.29 s = _

Σ(x − x )2

n − 1 = _

485.43

7 − 1 = 8.9947

INTERSTATE 75

x =

Σx

n =

472

8 = 59.00 s = _

Σ(x − x)2

n − 1 = _

134

8 − 1 = 4.3753

There is more variation, as measured by the standard deviation, in the U.S. 25 route

than in the I-75 route. This is consistent with his knowledge of the two routes; the U.S.

25 route contains more stoplights, whereas I-75 is a limited-access interstate highway.

However, the I-75 route is several miles longer. It is important that the service

EXAMPLE

Lammers Limos offers limousine service

from Government Center in downtown

Toledo,

Ohio, to Metro Airport in Detroit.

Sean Lammers, president of the company,

is considering two routes. One is via U.S.

25 and the other via I-75. He wants to

study the time it takes to drive to the airport

using each route and then compare

the results. He collected the following

sample data, which is reported in minutes.

Using the .10 significance level, is there a

difference in the variation in the driving

times for the two routes?