The F-Distribution
Illustrative Problem
A manufacturer of soft drink machines in concerned about the
variance in the amount filled for soft drinks. It has sample
tested each of two machines with the following results:
We want to test to see if the variance of the new machine is
more than the variance of the present machine. Use α=.01.
We always set up the problems so F-Stat > 1.

The hypotheses are:
m
2
H0 :
2
m
 
H A :
2
m
 
2
p
p
2
m
 1 (The Ratio equals 1)
2
2
P

2
p
 1 (The Ratio is greater than 1)
2
The sample statistic is:
F  Stat 
sm
2
sp


0 .0018
 2.25
0 .0008
In order to find a p-value, we need a sampling distribution for
the sample statistic, the sample variance.

Chapter 12
1
The F-Distribution
4.
Each sample must be from a normal distribution
Chapter 12
2
Finding the P-Value
We now have the F-Distribution, with an F-Stat of
2.25, df for the numerator of 24, and df for the
denominator of 21. We will use the Add-In Program
FDIST.
F Distribution (24, 21)
P-Value = 0.0323
F = 2.25
PRGM – FDIST
LOWER BOUND: 2.25
UPPER BOUND: 2ND E99
df NUMERATOR: 24
df DENOMINATOR: 21
OUTPUT: P-VALUE = 0.0323
Chapter 12
3
Illustrative Problem
m
2
H0 :
2
m
 
H A :
2
m
 
2
p
(24, 21)
p
2
P-value = 0.0323
m
2

2
P
p
2
F=2.25
Solving using TI-83 Black Box Program
STAT-TESTS-E:2SampFTest
Input: Stats
Sx1: 0.0018 (Problem gives s2, TI-83 requires Sx)
n1: 25
Sx2: 0.0008
n2: 22

σ1 > σ2
P-Value = 0.0323; F = 2.25

Ho cannot be rejected at α=0.01.
Chapter 12
4
Problem
Chapter 12
5
Problems
Chapter 12
6
Problem
Set up the problem so that the the F-Stat >1.
a. State the necessary hypotheses.
b. Find the p-value and state your conclusion.
c. What is the name of the model used for the
sampling distribution?
Chapter 12
7
Analysis of Variance
The analysis of variance process (ANOVA) will be
used to test hypotheses about several means.
We have previously looked at the Goodness of Fit
process that tests several proportions. We have
looked at hypotheses tests for one mean, two means,
and now we examine several means. The (ANOVA)
process requires the F- Distribution for the sampling
distribution.
Conditions required for (ANOVA) is that:
1. Samples are random and independent of each
other.
2. The effect of untested factors is normally
distributed and has constant variance.
The typical hypotheses is:
H 0 : 1   2   3   4   5
H A : At least one of means is different

Chapter 12
8
Illustrative Problem
The temperature at which a manufacturing process is believed
to affect the productivity of the process. Three different
production samples were taken when the temperature variable
(factor) is at three different temperatures. Following is the data:
The three means appear to be different based on the
samples. We want to test to see if that is a real
difference or a difference solely due to sampling
variation. We will assume the necessary conditions for
ANOVA are met.
The hypotheses are:
H 0 : 1   2   3
H A : At least one of the means is different

Chapter 12
9
ILLUSTRATIVE PROBLEM
BOX PLOT DISPALY
To get a picture of these three distributions, we make
side-by-side box plots.
On the TI-84, we input the data into Lists 1, 2, and 3.
We then go to 2nd Statplot and and set up a box plot
for list 1 in Plot1, for list 2 in Plot2 and for list 3 in Plot
3. We select ZoomStat.
Plot 1 68°
Plot 2 72°
Plot 3 76°
Chapter 12
10
Illustrative Problem: F Stat
In order to find a p-value from the F-Distribution, we need to
develop an F Statistic. We will do this by developing a ratio of
two sum of squares calculations. Recall that the formula for the
variance is: s 2  ( x  x ) 2 /( n  1) . We use the numerator to develop
the sum of the squares.
SS(Total)
is the sum of the squares if the three sets of data were

use to develop one grand mean. SS(Total) = 94.
Since there are thirteen numbers, the degrees of freedom for this
calculation is 13-1=12.
SS(Factor), SS(Temperature) is this problem, is the sum of the
squares for each of the three samples, added together.
SS(Treatment) = 84.5. Since there are three samples, the
degrees of freedom for this calculation is 3-1 = 2.
SS(Error) is SS(Total) – SS(Factor) = 94 – 84.5 = 9.5. The
degrees of freedom for this calculation is 12 – 2 = 10.
Mean Square for Factor: MS (Factor ) 
(Variation between sample means)
Mean Square for Error: MS ( Error
(Variation withinsamples)
)
SS (Factor )

df (Factor )
SS ( Error )
df ( Error )

9.5
84 .5
 42 .25
2
 .95
10

Chapter 12
11
Illustrative Problem: F Stat
The resulting ANOVA table for this problem is as follows:
F  Stat 
Variatin
Variation

The F Stat is
Between
Samples
Within Sampes
MS (Temperature )
MS ( Error )

42 .25
 44 .47
0.95
= 0.00001

Since the p-value is less than α=.05, we reject Ho
and conclude that at least one of the means is
different. Follow up analysis would be needed to
determine which one.
Chapter 12
12
Illustrative Problem- TI-83
Data in Lists
H 0 : 1   2   3
H A : At least one of the means is different

STAT – Edit (Enter Data in L1, L2, L3)
PRGM – ANOVA2
NUM LISTS? 3
(Enter the three lists)
Output: P-VALUE = 1.0543E-5; F = 44.4737;
MSF = 42.25; MSE=.95
Chapter 12
13
Illustrative Problem- TI-83
Statistics Input
An experiment was done with three measured treatments.
The statistics are given below:
2nd Matrix (Enter the data in matrix A as shown)
PRGM-ANOVA2
2: STATS IN MATX
ENTER
1 (CONTINUE)
ENTER
OUTPUT: P-Value = 2.2793 E-4: F=14.1109
Chapter 12
14
Problems
Chapter 12
15
Problems
Chapter 12
16
Problems
a.
b.
c.
d.
State the necessary hypotheses.
Sketch the side-by-side box plots. Does it appear
that the means are all the same?
Find the p-value and state your conclusion.
What is the name of the model used for the
sampling distribution?
Chapter 12
17
Problems
Sample
Size
Sample Sample
Mean
St. Dev.
Atlanta
6
24.67
7.76
Boston
7
33.00
9.56
Dallas
7
30.86
7.58
Philadelphia
5
32.20
7.47
Seattle
5
27.40
9.40
St. Louis
6
25.83
10.03
a. Test the hypotheses that not all the mean commute
times are all the same. State the appropriate
hypothesis.
b. Find the p-value and state your conclusion.
c. What is the name of the sampling distribution?
d. What is the F-Statistic, the df numerator and df
denominator?
Chapter 12
18