Islamic University of Gaza Statistics and Probability for

Islamic University of Gaza Statistics and Probability for Engineers (ENGC 6310) Lecture 8: One-Way ANOVA Prof. Dr. Yunes Mogheir Civil and Environmental Engineering Dept. First Semester/2019

Overview and One-Way ANOVA Slid Copyright © 2007 Pearson Education, Inc Publishing as Pearson

Overview v. Analysis of variance (ANOVA) is a method for testing the hypothesis that three or more population means are equal. v. For example: H 0: µ 1 = µ 2 = µ 3 =. . . µ k H 1: At least one mean is different Slid Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

F - distribution Slid Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

One-Way ANOVA An Approach to Understanding ANOVA 1. Understand that a small P-value (such as 0. 05 or less) leads to rejection of the null hypothesis of equal means. With a large P-value (such as greater than 0. 05), fail to reject the null hypothesis of equal means. 2. Develop an understanding of the underlying rationale by studying the examples in this section. Slid Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

One-Way ANOVA An Approach to Understanding ANOVA 3. Become acquainted with the nature of the SS (sum of squares) and MS (mean square) values and their role in determining the F test statistic. Slid Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

One-Way ANOVA Requirements 1. The populations have approximately normal distributions. 2. The populations have the same variance (or standard deviation ). 2 3. The samples are simple random samples. 4. The samples are independent of each other. Slid Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Procedure for testing Ho: µ 1 = µ 2 = µ 3 =. . . 1. Use Minitab or Excel. 2. Identify the P-value 3. Form a conclusion based on these criteria: If P-value , reject the null hypothesis of equal means. If P-value > , fail to reject the null hypothesis of equal means. Slid Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Example: Weights of Poplar Trees ( )ﺷﺠﺮ ﺍﻟﺼﻔﺼﺎﻑ Do the samples come from populations

Example: Weights of Poplar Trees Do the samples come from populations with different means? H 0: 1 = 2 = 3 = 4 H 1: At least one of the means is different from the others. For a significance level of = 0. 05, use Minitab or Excel, to test the claim that the four samples come from populations with means that are not all the same. Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Example: Weights of Poplar Trees Do the samples come from populations with different means? H 0: 1 = 2 = 3 = 4 H 1: At least one of the means is different from the others. The P-value of approximately 0. 007. Because the P-value is less than the significance level of = 0. 05, we reject the null hypothesis of equal means. There is sufficient evidence to support the claim that the four population means are not all the same. We conclude that those weights come from populations having means that are not all the same. Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

ANOVA Fundamental Concepts Test Statistic for One-Way ANOVA F= variance between samples variance within samples An excessively large F test statistic is evidence against equal population means. Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Relationships Between the F Test Statistic and P-Value Figure 12 -2 Slide Copyright ©

Calculations with Equal Sample Sizes v. Variance between samples = nsx 2 where sx 2 = variance of sample means v. Variance within samples = sp 2 where sp = pooled variance (or the mean of the sample variances) 2 Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Example: Sample Calculations Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Key Components of the ANOVA Method SS(total), or total sum of squares, is a measure of the total variation (around x) in all the sample data combined. SS(total) = (x – x) 2 Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Key Components of the ANOVA Method SS (treatment), also referred to as SS(factor) or SS(between groups) or SS(between samples), is a measure of the variation between the sample means. SS(treatment) = n 1(x 1 – x)2 + n 2(x 2 – x)2 +. . . nk(xk – x)2 = ni(xi - x)2 Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Key Components of the ANOVA Method SS(error), (also referred to as SS(within groups) or SS(within samples), is a sum of squares representing the variability that is assumed to be common to all the populations being considered. SS(error) = (n 1 – 1)s 12 + (n 2 – 1)s 22 + (n 3 – 1)s 32. . . nk(xk – 1)si 2 = (ni – 1)s 2 i Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Key Components of the ANOVA Method Given the previous expressions for SS(total), SS(treatment), and SS(error), the following relationship will always hold. SS(total) = SS(treatment) + SS(error) Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Mean Squares (MS) MS(treatment) is a mean square for treatment, obtained as follows: MS(treatment) = SS (treatment) k – 1 MS(error) is a mean square for error, obtained as follows: MS(error) = SS (error) N – k Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Mean Squares (MS) MS(total) is a mean square for the total variation, obtained as follows: MS(total) = SS(total) N – 1 Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Test Statistic for ANOVA with Unequal Sample Sizes F= MS (treatment) MS (error) v Numerator df = k – 1 v Denominator df = N – k Slide Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.