Bennie Waller wallerbd@longwood 434-395-2046

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Bennie Waller wallerbd@longwood.edu 434-395-2046 Longwood University 201 High Street Farmville, VA 23901. Bennie Waller wallerbd@longwood.edu 434-395-2046 Longwood University 201 High Street Farmville, VA 23901. The F Distribution. It is

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Bennie Waller

wallerbd@longwood.edu

434-395-2046

Longwood University201 High StreetFarmville, VA 23901

Bennie Waller

wallerbd@longwood.edu

434-395-2046

Longwood University201 High StreetFarmville, VA 23901

The F Distribution

• It is
• used to test whether two samples have equal variances
• used to compare the means of more than two populations simultaneously. The simultaneous comparison of several population means is called analysis of variance(ANOVA).

The F Distribution

H0: µ1 = µ2 =…= µk

H1: The means are not all equal

• H0: σ12 = σ22
• H1: σ12≠σ22

ANOVA

Test for Equal Variances - Example

Step 1: The hypotheses are:

H0: σ12 = σ22

H1: σ12≠σ22

Step 2: The significance level is .10.

Step 3: The test statistic is the F distribution.

12-5

ANOVA

Test for Equal Variances - Example

Step 4: State the decision rule.

Reject H0 ifF > F/2,v1,v2

F > F.10/2,7-1,8-1

F > F.05,6,7

12-6

See example in spreadsheet “VARanova” tab

B

A

ANOVA

Comparing Means of Two or More Populations

• TheNull Hypothesis is that the population means are all the same. The Alternative Hypothesis is that at least one of the means is different.
• TheTest Statistic is the F distribution.
• TheDecision rule is to reject the null hypothesis if F (computed) is greater than F (table) with numerator and denominator degrees of freedom.
• Hypothesis Setup and Decision Rule:

H0: µ1 = µ2 =…= µk

H1: The means are not all equal

Reject H0 if F > F,k-1,n-k

12-9

ANOVA

Analysis of Variance – F statistic
• If there are k populations being sampled, the numerator degrees of freedom is k – 1.
• If there are a total of n observations the denominator degrees of freedom is n – k.
• The test statistic is computed by:

12-10

Consider the following example comparing the difference in means of three groups

NOTICE THE DIFFERENCE IN VARIATION ACROSS GROUPS RELATIVE TO WITHIN GROUPS

See example in spreadsheet “ANOVAMeans” tab

Problem: A company compared the variance of salaries for employees who have been employed for 5 years or less with employees who have been employed for 10 years or more. They randomly selected 21 employees with 5 years or less experience and 15 employees with 10 years or more experience. The standard deviation for the group with 5 years or less experience was \$2,225; the standard deviation for the group with 10 years or more experience is \$1,875. Using the 0.05 significance level, what is the decision regarding the null hypothesis?

44. Given the following Analysis of Variance table for three treatments each with six observations.

Given the following Analysis of Variance table for three treatments each with six observations. What is the decision regarding the null hypothesis?