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Chapter 12. Analysis of Variance. Goals. List the characteristics of the F distribution Conduct a test of hypothesis to determine whether the variances of two populations are equal Discuss the general idea of analysis of variance Organize data into a ANOVA table

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Chapter 12

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Chapter 12 l.jpg

Chapter 12

Analysis of Variance


Goals l.jpg

Goals

  • List the characteristics of the F distribution

  • Conduct a test of hypothesis to determine whether the variances of two populations are equal

  • Discuss the general idea of analysis of variance

  • Organize data into a ANOVA table

  • Conduct a test of hypothesis among three or more treatment means


F distribution l.jpg

F Distribution

  • Used to test whether two samples are from populations having equal variances

  • Applied when we want to compare several population means simultaneously to determine if they came from equal population

    • ANOVA

      • Analysis of variance

  • In both situations:

    • Populations must be Normally distributed

    • Data must be Interval-scale or higher


Characteristics of the f distribution l.jpg

Characteristics Of The F Distribution

  • Family of F Distributions

    • Each is determined by:

      • df in numerator

        • comes from pop. 1 which has larger sample variation

      • df in denominator

        • comes from pop. 2 which has smaller sample variation

  • F distribution is continuous

    • F Value can assume an infinite number of values from 0 to ∞

  • Value for F Distribution cannot be negative

    • Smallest value = 0

  • Positively skewed

    • Long tail is always to right

    • As # of df increases in both the numerator and the denominator, the distribution approaches normal

  • Asymptotic

    • As X increases the F curve approaches the X-axis


Why do we want to compare to see if two population have equal variances l.jpg

What if two machines are making the same part for an airplane?

Do we want the parts to be identical or nearly identical? Yes!

We would test to see if the means are the same: Chapter 10 & 11

We would test to see if the variation is the same for the two machines: Chapter 12

What if two stocks have similar mean returns?

Would we like to test and see if one stock has more variation than the other?

Why Do We Want To Compare To See If Two Population Have Equal Variances?


Why do we want to compare to see if two population have equal variances6 l.jpg

Why Do We Want To Compare To See If Two Population Have Equal Variances?

  • Remember Chapter 11: Assumptions for small sample tests of means:

    • Sample populations must follow the normal distribution

    • Two samples must be from independent (unrelated) populations

    • The variances & standard deviations of the two populations are equal


Conduct a test of hypothesis to determine whether the variances of two populations are equal l.jpg

Always list the sample

with the

larger sample variance

as population 1

(allows us to

use fewer

tables)

Conduct A Test Of Hypothesis To Determine Whether The Variances Of Two Populations Are Equal

  • To conduct a test:

  • Conduct two random samples

  • List population 1 as the sample with the largest variance:

    • n1 = # of observations

    • s1^2 = sample variance

    • n1 – 1 = df1 = degree of freedom (numerator for critical value lookup)

  • List population 2 as the sample with the smaller variance:

    • n2 = # of observations

    • s2^2 = sample variance

    • n2 – 1 = df2 = degree of freedom (denominator for critical value lookup)


Step 1 state null and alternate hypotheses l.jpg

Step 1:State null and alternate hypotheses

  • List the population with the suspected largest variance as population 1

    • Because we want to limit the number of F tables we need to use to look up values, we always put the larger variance in the numerator and the smaller variance in the denominator

    • This will force the F value to be at least 1

    • We will only use the right tail of the F distribution

  • Examples of Step 1:


Step 2 select a level of significance l.jpg

Step 2: Select a level of significance:

  • Appendix G only lists significance levels: .05 and .01

Significance level = .10

.10/2 = .05

Use .05 table in

Appendix G

Significance level = .05

Use .05 table in

Appendix G


Step 3 identify the test statistic f find critical value and draw picture l.jpg

Step 3: Identify the test statistic (F), find critical value and draw picture

If you have a df

that is not listed

in the border,

calculate your F

by estimating

a value

between

two values.

HW #5:

df = 11,

use value

Between

10 & 12

Book says:

(3.14+3.07)/2 =

3.105  3.10

  • Look up Critical value in Appendix G and draw your picture


Step 4 l.jpg

Step 4

  • Step 4: Formulate a decision rule:

    • Example:

    • If our calculated test statistic is greater than 3.87, reject Ho and accept H1, otherwise fail to reject Ho


Step 5 l.jpg

Larger variancein numerator,

always!!

Step 5

  • Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses

  • Test Statistic F:

  • Example Conclusion for a two tail test:

    • Fail to reject null

      • “The evidence suggests that there is not a difference in variation”

    • Reject null and accept alternate

      • “The evidence suggests that there is a difference in variation”

Let’s Look at

Handout 


Example 1 l.jpg

Colin, a stockbroker at Critical Securities, reported that the mean rate of return on a sample of 10 software stocks was 12.6 percent with a standard deviation of 3.9 percent.

The mean rate of return on a sample of 8 utility stocks was 10.9 percent with a standard deviation of 3.5 percent. At the .05 significance level, can Colin conclude that there is more variation in the software stocks?

Example 1


Example 1 continued l.jpg

Step 1: The hypotheses are

Step 2: The significance level is .05.

Step 3: The test statistic is the F distribution.

Example 1 continued


Example 1 continued15 l.jpg

Step 4:H0 is rejected if F>3.68 or if p < .05. The degrees of freedom are n1-1 or 9 in the numerator and n1-1 or 7 in the denominator.

Step 5: The value of F is computed as follows.

H0 is not rejected. There is insufficient evidence to show more variation in the software stocks.

Example 1 continued


Anova analysis of variance l.jpg

ANOVAAnalysis Of Variance

  • Technique in which we compare three or more population means to determine whether they could be equal

    • Assumptions necessary:

      • Populations follow the normal distribution

      • Populations have equal standard deviations ()

      • Populations are independent

  • Why ANOVA?

    • Using t-distribution leads to build up of type 1 error

  • “Treatment” = different populations being examined


Case where treatment means are different l.jpg

Case Where Treatment Means Are Different


Case where treatment means are the same l.jpg

Case Where Treatment Means Are The Same


Example of anova test to see if four treatment means are the equal l.jpg

Example Of ANOVA Test To See If Four Treatment Means Are The Equal

  • 22 students earned the following grades in Professor Rad’s class. The grades are listed under the classification the student gave to the instructor

  • Is there a difference in the mean score of the students in each of the four categories?

  • Use significance level α = .01


Conduct a test of hypothesis among four treatment means l.jpg

Conduct A Test Of Hypothesis Among Four Treatment Means

  • Step 1: State H0 and H1

    H0 : µ1 = µ2 = µ3 = µ4

    H1 : The Mean scores are not all equal (at least one treatment mean is different)

  • Step 2: Significance Level?

    α = .01


Step 3 determine test statistic and select critical value l.jpg

Step 3: Determine Test Statistic And Select Critical Value


Step 4 state decision rule l.jpg

Step 4: State Decision Rule

  • If our calculated test statistic is greater than we reject H0 and accept H1, otherwise we fail to reject H0

Now we move on to Step 5: Select the sample, perform calculations, and make a decision…

Are you ready for a lot of procedures?!!


Slide23 l.jpg

  • The idea is: If we estimate variation in two ways and use one estimate in the numerator and the other estimate in the denominator:

    • If we divide and get 1 or close to 1, the sample means are assumed to be the same

    • If we get a number far from 1, we say that the means are assumed to be different

  • The F critical value will determined whether we are close to 1 or not


Anova table so far l.jpg

ANOVA Table So Far

Let’s go calculate this!


Calculation 1 treatment means and overall mean l.jpg

Calculation 1: Treatment Means and Overall Mean


Calculation 2 total variation l.jpg

Calculation 2: Total Variation


Calculation 2 total variation27 l.jpg

Calculation 2: Total Variation


Anova table so far28 l.jpg

ANOVA Table So Far

Let’s go calculate this!


Calculation 3 random variation l.jpg

Calculation 3: Random Variation


Calculation 3 random variation30 l.jpg

Calculation 3: Random Variation


Anova table so far31 l.jpg

ANOVA Table So Far

Let’s go calculate this!


Calculation 4 treatment variation l.jpg

Calculation 4: Treatment Variation

Simple Subtraction!


Calculation 5 mean square estimate of variation l.jpg

Calculation 5: Mean Square (Estimate of Variation)


Calculation 6 f l.jpg

Calculation 6: F


Step 5 make a decision l.jpg

Step 5: Make A Decision

  • Because is less than 5.09, we fail to reject H0

  • The evidence suggests that the mean score of the students in each of the four categories are equal (no difference)


Summarize chapter 12 l.jpg

Summarize Chapter 12

  • List the characteristics of the F distribution

  • Conduct a test of hypothesis to determine whether the variances of two populations are equal

  • Discuss the general idea of analysis of variance

  • Organize data into a ANOVA table

  • Conduct a test of hypothesis among three or more treatment means


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