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Chapter 13 Comparing Three or More Means Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Section 13.1 Comparing Three or More Means (One-Way Analysis of Variance) Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Analysis of Variance (ANOVA) is an inferential method used to test the equality of three or more population means. 13-3 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. CAUTION! Do not test H0: μ1 = μ2 = μ3 by conducting three separate hypothesis tests, because the probability of making a Type I error will be much higher than α. 13-4 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Requirements of a One-Way ANOVA Test 1. There must be k simple random samples; one from each of k populations or a randomized experiment with k treatments. 2. The k samples are independent of each other; that is, the subjects in one group cannot be related in any way to subjects in a second group. 3. The populations are normally distributed. 4. The populations must have the same variance; that is, each treatment group has the population variance σ2. 13-5 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Testing a Hypothesis Regarding k = 3 H0: μ1 = μ2 = μ3 H1: At least one population mean is different from the others 13-6 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Testing Using ANOVA The methods of one-way ANOVA are robust, so small departures from the normality requirement will not significantly affect the results of the procedure. In addition, the requirement of equal population variances does not need to be strictly adhered to, especially if the sample size for each treatment group is the same. Therefore, it is worthwhile to design an experiment in which the samples from the populations are roughly equal in size. 13-7 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Verifying the Requirement of Equal Population Variance The one-way ANOVA procedures may be used if the largest sample standard deviation is no more than twice the smallest sample standard deviation. 13-8 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Parallel Example 1: Verifying the Requirements of ANOVA The following data represent the weight (in grams) of pennies minted at the Denver mint in 1990,1995, and 2000. Verify that the requirements in order to perform a one-way ANOVA are satisfied. 13-9 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. 1990 2.50 2.50 2.49 2.53 2.46 2.50 2.47 2.53 2.51 2.49 2.48 13-10 1995 2.52 2.54 2.50 2.48 2.52 2.50 2.49 2.53 2.48 2.55 2.49 2000 2.50 2.48 2.49 2.50 2.48 2.52 2.51 2.49 2.51 2.50 2.52 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Solution 1. The 3 samples are simple random samples. 2. The samples were obtained independently. 3. Normal probability plots for the 3 years follow. All of the plots are roughly linear so the normality assumption is satisfied. 13-11 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Solution 4. The sample standard deviations are computed for each sample using Minitab and shown on the following slide. The largest standard deviation is not more than twice the smallest standard deviation (2•0.0141 = 0.0282 > 0.02430) so the requirement of equal population variances is considered satisfied. 13-12 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Descriptive Statistics Variable 1990 1995 2000 N 11 11 11 Variable 1990 1995 2000 Minimum 2.4600 2.4800 2.4800 13-13 Mean 2.4964 2.5091 2.5000 Median 2.5000 2.5000 2.5000 Maximum 2.5300 2.5500 2.5200 TrMean 2.4967 2.5078 2.5000 Q1 2.4800 2.4900 2.4900 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. StDev 0.0220 0.0243 0.0141 Q3 2.5100 2.5300 2.5100 SE Mean 0.0066 0.0073 0.0043 The basic idea in one-way ANOVA is to determine if the sample data could come from populations with the same mean, μ, or suggests that at least one sample comes from a population whose mean is different from the others. To make this decision, we compare the variability among the sample means to the variability within each sample. 13-14 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. We call the variability among the sample means the between-sample variability, and the variability of each sample the within-sample variability. If the between-sample variability is large relative to the within-sample variability, we have evidence to suggest that the samples come from populations with different means. 13-15 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. ANOVA F-Test Statistic The analysis of variance F-test statistic is given by between - sample variability F0 = within - sample variability mean square due to treatments = mean square due to error MST = MSE 13-16 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Computing the F-Test Statistic Step 1: Compute the sample mean of the combined data set by adding up all the observations and dividing by the number of observations. Call this value x . Step 2: Find the sample mean for each sample (or treatment). Let x1 represent the sample mean of sample 1, x2 represent the sample mean of sample 2, and so on. Step 3: Find the sample variance for each sample (or treatment). Let s12 represent the sample variance for sample 1, s22 represent the sample variance for sample 2, and so on. 13-17 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Computing the F-Test Statistic Step 4: Compute the sum of squares due to treatments, SST, and the sum of squares due to error, SSE. Step 5: Divide each sum of squares by its corresponding degrees of freedom (k – 1 and n – k, respectively) to obtain the mean squares MST and MSE. Step 6: Compute the F-test statistic: mean square due to treatments MST F0 mean square due to error MSE 13-18 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Solution The results of the computations for the data that led to the F-test statistic are presented in the following table. ANOVA Table: Source of Sum of Degrees of Variation Squares Freedom Treatment Mean Squares 0.0009 2 0.0005 Error 0.013 30 0.0004 Total 0.0139 32 13-19 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. F-Test Statistic 1.25 Decision Rule in the One-Way ANOVA Test If the P-value is less than the level of significance, α, reject the null hypothesis. 13-20 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Example Compute the F-test statistic for the following data. 𝒙𝟏 𝒙𝟐 𝒙𝟑 4 7 10 5 8 10 6 9 11 6 7 11 4 9 13 13-21 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Example Suppose the teeth researcher wans to determine if there is a difference in the mean shear bond strength among the four treatment groups at the 𝛼 = 0.05 level of significance. 13-22 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Example The data in the table represent the number of pods on a random sample of soybean plants for various plot types. An agricultural researcher wants to determine if the mean numbers of pods for each plot type are equal. Use the 𝛼 = 0.05 level of significance. 13-23 Copyright © 2017, 2013, and 2010 Pearson Education, Inc. Plot Type Pods Liberty 32 31 36 35 41 34 39 37 38 No till 34 30 31 27 40 33 37 42 39 Chisel plowed 34 37 24 23 32 33 27 34 30 13-24 Copyright © 2017, 2013, and 2010 Pearson Education, Inc.