## CHAPTER 12

### THE

*F*

### DISTRIBUTION

**Definition **

1._{ } The F* distribution* is continuous and skewed to the right.
2._{ } The F* distribution* has two numbers of degrees of

freedom: *df*_{ for the numerator and }*df*_{ for the denominator. }

### Example 12-1

Find the *F*_{ value for 8 degrees of freedom for the numerator, }

### Figure 12.2 The critical value of

*F*

### for 8

*df*

### for the numerator, 14

*df*

### ONE-WAY ANALYSIS OF VARIANCE

Calculating the Value of the Test Statistic

### ONE-WAY ANALYSIS OF VARIANCE

**Definition **

**ANOVA is a procedure used to test the null hypothesis that **

### Assumptions of One-Way ANOVA

The following assumptions must hold true to use one-way

**ANOVA. **

1. The populations from which the samples are drawn are (approximately) normally distributed.

2. The populations from which the samples are drawn have the same variance (or standard deviation).

### Calculating the Value of the Test Statistic

**Test Statistic F for a One-Way ANOVA Test **

The value of the test statistic * F* for an ANOVA test is

calculated as

The calculation of MSB and MSW is explained in Example

12-2.

**Variance between samples**

**MSB**

** or **

**Variance within samples**

**MSW**

### Example 12-2

### Example 12-2

Calculate the value of the test statistic *F*. Assume that all the

### Example 12-2: Solution

Let

*x* = the score of a student

*k* = the number of different samples (or treatments)

### Example 12-2: Solution

To calculate MSB and MSW, we first compute the

### Between- and Within-Samples Sums of Squares

The between-samples sum of squares, denoted by **SSB**,

### Between- and Within-Samples Sums of Squares

The within-samples sum of squares, denoted by **SSW**, is

### Calculating the Values of MSB and MSW

**MSB** and **MSW** are calculated as

where *k* – 1 and *n* – *k* are, respectively, the *df* for the

numerator and the *df*_{ for the denominator for the }_{F}

distribution. Remember, *k*_{ is the number of different }

### Example 12-3

Reconsider Example 12-2 about the scores of 15

fourth-grade students who were randomly assigned to three groups in order to experiment with three different methods of

teaching arithmetic. At the 1% significance level, can we reject the null hypothesis that the mean arithmetic score of all fourth-grade students taught by each of these three

methods is the same? Assume that all the assumptions

### Example 12-3: Solution

Because we are comparing the means for three normally

distributed populations, we use the *F*_{ distribution to make this }

### Example 12-3: Solution

**Step 4 & 5: **

The value of the test statistic *F* = 1.09

It is less than the critical value of *F* = 6.93

It falls in the nonrejection region

Hence, we fail to reject the null hypothesis.

### Example 12-4

From time to time, unknown to its employees, the research department at Post Bank observes various employees for their work productivity. Recently this department wanted to check whether the four tellers at a branch of this bank serve, on average, the same number of customers per hour. The research manager observed each of the four tellers for a

certain number of hours. The following table gives the number of customers served by the four tellers during each of the

### Example 12-4

At the 5% significance level, test the null hypothesis that the

mean number of customers served per hour by each of these four tellers is the same. Assume that all the assumptions

### Example 12-4: Solution

**Step 2: **

Because we are testing for the equality of four means for
four normally distributed populations, we use the *F*

### Example 12-4: Solution

**Step 5: **

The value for the test statistic *F* = 9.69

It is greater than the critical value of *F* = 3.16

It falls in the rejection region

Consequently, we reject the null hypothesis