The F-DistributionIllustrative ProblemA manufacturer of soft drink machines is 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 set up the problem so F-Stat > 1.The hypotheses are:

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H0 :σ m2 =σ p

2⇒σ m2

σ p2=1 (The Ratio equals 1)

HA :σ m2 >σ P

2⇒σ m2

σ p2>1 (The Ratio is greater than 1)

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The sample statistic is:

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F − Stat =sm2

sp2=0.0018

0.0008=2.25

In order to find a p-value, we need a sampling distribution for the sample statistic, the ratio of the sample variances.

1Sec 10.5, Page 226

The F-Distribution

2Sec 10.5, Page 226

Each sample must be from a normal distribution4.

Finding the P-ValueWe 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 = 2.25

PRGM – FDISTLOWER BOUND: 2.25UPPER BOUND: 2ND E99df NUMERATOR: 24df DENOMINATOR: 21OUTPUT: P-VALUE = 0.0323

F Distribution (24, 21)

3Sec 10.5, Page 226

P-Value = 0.0323

Illustrative Problem

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H0 :σ m2 =σ p

2⇒σ m2

σ p2 =1

HA :σ m2 >σ P

2⇒σ m2

σ p2 >1

Solving using TI-83 Black Box ProgramSTAT-TESTS-E:2SampFTestInput: StatsSx1: (Problem gives s2, TI-83 requires Sx)n1: 25Sx2: n2: 22σ1 > σ2P-Value = 0.0323; F = 2.25Ho cannot be rejected at α=0.01.

(24, 21)

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0.0018

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0.0008

F=2.25

P-value = 0.0323

4Sec 10.5, Page 226

Problem

5Problems, Page 234

Problems

6Problems, Page 234

Problem

7Problems, Page 234

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?

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 populations are normal

The typical hypotheses is:

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H0 :μ1 = μ2 = μ3 = μ4 = μ5HA : At least oneof means is different

8Sec 12.1, Page 254

Illustrative ProblemThe 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:

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H0 :μ1 = μ2 = μ3HA : At least one of the means is different

9Sec 1.1, Page 256

ILLUSTRATIVE PROBLEMBOX 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°

10Sec 12.1, Page 256

Illustrative Problem: F StatIn 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: . 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. The grand mean is

Since there are thirteen numbers, the degrees of freedom for this calculation is 13-1=12. SS(Factor), SS(Temperature) is the weighted sum of the squares of each sample mean from the grand mean.

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.

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s2 = Σ(x − x )2 /(n −1)

11Sec 12.1, Page 258

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SS(Total) = (10 − 7)2 + (12 − 7)2 + ...+ (5 − 7)2 + (4 − 7)2 = 94

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SS(Factor) = n1(x 1 − x )2 + n2(x 2 − x )2 + n3(x 3 − x )2

= 4(10.25 − 7)2 +5(7 − 7)2 + 4(3.75 − 7)2 = 84.50

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x = (10 +12 +10 +9 + 7 +6 + 7 +8 + 7 + 3+ 3+5 + 4) /13 = 7

Illustrative Problem: F StatThe resulting ANOVA table for this problem is as follows:

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The F Stat isMS(Temperature )

MS(Error)=42.25

0.95= 44.47

= 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.

12Sec 12.1, Page 258

Illustrative Problem- TI-83Data in Lists

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H0 :μ1 = μ2 = μ3HA : At least one of the means is different

STAT – Edit (Enter Data in L1, L2, L3)PRGM – ANOVA2NUM LISTS? 3(Enter the three lists)Output: P-VALUE = 1.0543E-5; F = 44.4737;MSF = 42.25; MSE=.95

13Sec 12.1, Page 258

Illustrative Problem- TI-83Statistics 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-ANOVA22: STATS IN MATXENTER1 (CONTINUE)ENTEROUTPUT: P-Value = 2.2793 E-4: F-Ratio =12.1378

14Chapter 12

Problems

15Sec 12.1, Page 267

Problems

16Problems, Page 270

Problems

17Sec 12.1, Page 268

a. State the necessary hypotheses.b. Sketch the side-by-side box plots. Does it appear

that the means are all the same?c. Find the p-value and state your conclusion.d. What is the name of the model used for the

sampling distribution?

Problems

Problems, Page 268 18

SampleSize

SampleMean

SampleSt. 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?