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Now that the degrees of freedom and sum of squares column values have been determined, we can calculate the mean square for treatments, MSTr, and mean square for error, MSE. Calculate these values and place them in the ANOVA table, rounding each to three decimal places

Now that the degrees of freedom and sum of squares column values have been determined, we can calculate the mean square for treatments, MSTr, and mean square for error, MSE. Calculate these values and place them in the ANOVA table, rounding each to three decimal places

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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The values in the column for the degrees of freedom are based on the number of k population means, or treatments, and the total number of observations, N. In the sum of squares column, SSTT represents a measure of differences among the sample means. The value of SSE is a measure of variability within the k samples. Note that the values in the mean square column are calculated based on the sum of squares and degrees of freedom for the treatments row and error row. These mean square values are then used to calculate the F test statistic. The P-value is based on the F test statistic with k - 1 numerator degrees of freedom and N - k denominator degrees of freedom. The given data are below. Treatment 1 Treatment 2 Treatment 3 Treatment 4 9 5 5 6. 3 1 14 7 4 2 1 4. 3 12 3 4 9 2 2 9 1 2 14 8 3 3 3 3 3 4. 11 11 7 5 10 1 10
Transcribed Image Text:The values in the column for the degrees of freedom are based on the number of k population means, or treatments, and the total number of observations, N. In the sum of squares column, SSTT represents a measure of differences among the sample means. The value of SSE is a measure of variability within the k samples. Note that the values in the mean square column are calculated based on the sum of squares and degrees of freedom for the treatments row and error row. These mean square values are then used to calculate the F test statistic. The P-value is based on the F test statistic with k - 1 numerator degrees of freedom and N - k denominator degrees of freedom. The given data are below. Treatment 1 Treatment 2 Treatment 3 Treatment 4 9 5 5 6. 3 1 14 7 4 2 1 4. 3 12 3 4 9 2 2 9 1 2 14 8 3 3 3 3 3 4. 11 11 7 5 10 1 10
Step 4 The sample size, mean, and variance of each treatment as well as the overall sample mean are summarized below. "1= 18, x, = 6.1111, s,2 = 21.9869 n2 = 25, x, = 6.4, s,2 = 13.0000 n3 = 17, x3 = 3.6471, 5, - 6.2426 nA = 14, x- 2.7857, 5,2 = 4.6429 = = 5.0135 Use these values to calculate SSTT, rounding intermediate values to four decimal places and the result to three decimal places. STr = n,(x, - 7)2² + ng, - 7)² + = ng, - 7)2 + ng2 -? + ng3 - 2 + n4 - 52 = 18(5.1111 - 5.0135)? + 25(6.4 - 5.0135)2 + 17(3.6471 - 5.0135)2 + V + 14 2.7857 2.7857 5.0135 = 170.968 170.969 The next value to find is the sum of squares due to error, SSE. Use the above sample sizes and variances to calculate SSE, rounding intermediate values to four decimal places and the result to three decimal places. 2 SSE = (n, - 1)5, + (n2 - 1)5,2 + + (ng - 1)s, = (n, - 1)5,2 + (n, - 1)5,2 + (n3 - 1)5,2 + (n4- 1)5,2 = (18 - 1)(21.9869) + (25 - 1)(13.0000) + (17 - 1)(6.2426) + (14 - 1)4.6429 4.6429 = 846.017 846.017 Step 5 Now that the degrees of freedom and sum of squares column values have been determined, we can calculate the mean square for treatments, MSTT, and mean square for error, MSE. Calculate these values and place them in the ANOVA table, rounding each to three decimal places. Source of Variation Degrees of Freedom Sum of Squares Mean Square F SSTT MSTr = k - 1 k - 1 = 3 SSTT = 170.969 170.969 Treatments SSE N - k MSE = Error N - k = 70 SSE = 846.017 846.017
Transcribed Image Text:Step 4 The sample size, mean, and variance of each treatment as well as the overall sample mean are summarized below. "1= 18, x, = 6.1111, s,2 = 21.9869 n2 = 25, x, = 6.4, s,2 = 13.0000 n3 = 17, x3 = 3.6471, 5, - 6.2426 nA = 14, x- 2.7857, 5,2 = 4.6429 = = 5.0135 Use these values to calculate SSTT, rounding intermediate values to four decimal places and the result to three decimal places. STr = n,(x, - 7)2² + ng, - 7)² + = ng, - 7)2 + ng2 -? + ng3 - 2 + n4 - 52 = 18(5.1111 - 5.0135)? + 25(6.4 - 5.0135)2 + 17(3.6471 - 5.0135)2 + V + 14 2.7857 2.7857 5.0135 = 170.968 170.969 The next value to find is the sum of squares due to error, SSE. Use the above sample sizes and variances to calculate SSE, rounding intermediate values to four decimal places and the result to three decimal places. 2 SSE = (n, - 1)5, + (n2 - 1)5,2 + + (ng - 1)s, = (n, - 1)5,2 + (n, - 1)5,2 + (n3 - 1)5,2 + (n4- 1)5,2 = (18 - 1)(21.9869) + (25 - 1)(13.0000) + (17 - 1)(6.2426) + (14 - 1)4.6429 4.6429 = 846.017 846.017 Step 5 Now that the degrees of freedom and sum of squares column values have been determined, we can calculate the mean square for treatments, MSTT, and mean square for error, MSE. Calculate these values and place them in the ANOVA table, rounding each to three decimal places. Source of Variation Degrees of Freedom Sum of Squares Mean Square F SSTT MSTr = k - 1 k - 1 = 3 SSTT = 170.969 170.969 Treatments SSE N - k MSE = Error N - k = 70 SSE = 846.017 846.017
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