1. Bennie D Waller, Longwood University Statistics Bennie Waller wallerbd@longwood.edu 434-395-2046 Longwood University 201 High Street Farmville, VA 23901

2. Bennie D Waller, Longwood University Analysis of variance ANOVA Bennie Waller wallerbd@longwood.edu 434-395-2046 Longwood University 201 High Street Farmville, VA 23901

3. Bennie D Waller, Longwood University ANOVA 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

4. Bennie D Waller, Longwood University ANOVA The F Distribution H 0 : µ 1 = µ 2 =…= µ k H 1 : The means are not all equal H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2

5. Bennie D Waller, Longwood University Test for Equal Variances - Example Step 1: The hypotheses are: H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 Step 2: The significance level is.10. Step 3: The test statistic is the F distribution. 12-5 ANOVA

6. Bennie D Waller, Longwood University Step 4: State the decision rule. Reject H 0 if F > F  /2,v1,v2 F > F.10/2,7-1,8-1 F > F.05,6,7 Test for Equal Variances - Example 12-6 ANOVA

7. Bennie D Waller, Longwood University ANOVA Consider the following example of travel times on different routes F-Test Two-Sample for Variances Route-25I-75 Mean58.285759 Variance80.904619.1426 Observations78 df67 F4.22638 P(F<=f) one-tail0.04037 F Critical one-tail3.86599 See example in spreadsheet “VARanova” tab

8. Bennie D Waller, Longwood University Variance A B

9. Bennie D Waller, Longwood University The Null Hypothesis is that the population means are all the same. The Alternative Hypothesis is that at least one of the means is different. The Test Statistic is the F distribution. The Decision 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: Comparing Means of Two or More Populations H 0 : µ 1 = µ 2 =…= µ k H 1 : The means are not all equal Reject H 0 if F > F ,k-1,n-k 12-9 ANOVA

10. Bennie D Waller, Longwood University 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 ANOVA

11. Bennie D Waller, Longwood University ANOVA Number of orders processed AndyBettyCathy 556647 547651 596746 567148 µ=56µ=70µ=48

12. Bennie D Waller, Longwood University ANOVA Anova: Single Factor SUMMARY GroupsCountSumAverageVariance Andy4224564.666667 Betty42807020.66667 Cathy4192484.666667 ANOVA Source of VariationSSdfMSFP-valueF crit Between Groups992249649.61.38E-054.256495 Within Groups90910 Total108211 Consider the following example comparing the difference in means of three groups See example in spreadsheet “ANOVAMeans” tab NOTICE THE DIFFERENCE IN VARIATION ACROSS GROUPS RELATIVE TO WITHIN GROUPS

13. Bennie D Waller, Longwood University ANOVA 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?

14. Bennie D Waller, Longwood University ANOVA 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?

15. Bennie D Waller, Longwood University End