The following data are from a completely randomized design. a. Compute the sum of squares betwee

The following data are from a completely randomized design....

Key Concepts

F-Test in ANOVA

The F-test in ANOVA is used to compare the variance explained by the treatment effects (MSB) to the unexplained variance (MSE). The resulting F-statistic follows an F-distribution under the null hypothesis, which states that all group means are equal. A significant F-test result indicates that at least one treatment mean differs significantly from the others.

Completely Randomized Design

A completely randomized design is an experimental setup where subjects or experimental units are randomly assigned to different treatment groups. This design ensures that the groups are comparable and that the variability among observations is not systematically influenced by external factors, thereby allowing for an unbiased estimation of the treatment effects.

Sum of Squares Between Treatments

The sum of squares between treatments (also called the treatment sum of squares) measures the variability among the group means. It quantifies how much each group mean deviates from the overall mean, and it is used to assess whether the treatments have significantly different effects on the response variable.

Sum of Squares Due to Error

The sum of squares due to error (or within-group sum of squares) captures the variation within each treatment group that cannot be explained by the treatment effect. It represents the random error or residual variation and is critical for assessing the model's accuracy in partitioning total variability.

Mean Square

Mean squares are obtained by dividing the sum of squares by their corresponding degrees of freedom. In ANOVA, the mean square between treatments (MSB) and the mean square error (MSE) are compared to assess if the observed variability between groups is statistically significant compared to the within-group (error) variability.

ANOVA Table

The ANOVA table is a structured summary that partitions the total variation into components attributable to treatment effects and random error. It includes columns for degrees of freedom, sum of squares, mean squares, and the F-statistic. This table provides a framework for testing if there are significant differences among the treatment means.

Transcript

00:01 So in this problem, we're supposed to basically at the very end come up with a p value to, or a p value to calculate the significance and see if the means for these three treatment groups are equal.

00:16 And what this problem does really well is basically give you a step by step for how you should approach anova problems altogether.

00:22 So at the very end, we're supposed to create an nova table, which is this thing.

00:26 And this anova table will basically give you all the information.

00:30 You need in order to conduct this hypothesis test at the very end.

00:34 But in between, we need to come up with some values.

00:37 So here i have the given information in the problem.

00:41 The mean for the treatment group a is 156.

00:45 The mean for the treatment group b is 142.

00:47 And the mean for the treatment group c is 134.

00:50 Then the variance for the treatment group a is 164 .4.

00:54 Variance for the treatment group b is 131 .2.

00:57 And the variance for the treatment group c is 110 .4.

01:01 So the first thing the question asks us to do is find the sum of squares between treatments.

01:06 And the sum of squares between treatments is also known as the, or the first thing we need to find in order to calculate a sum of squares between treatments is what we call the grand mean, which is denoted by x double bar.

01:21 So that is just the sample mean of a minus plus the sample mean of b plus the sample mean of c all over the number of treatment groups, which we will call n.

01:38 Okay, so in this situation, it will be 156 plus 142 plus 134 over 3, which is equal to 144.

01:50 So this is our grand mean.

01:53 This value comes up a lot in in our innova tests or in our calculations for an an annova.

01:59 And now to actually calculate the sum of squares between treatments, we will take the sum, the sum from j equals 1 to k, where j is just a holder value, and k is the number of treatments.

02:22 I know we already used n.

02:23 Actually, let me just use n.

02:25 So n is the number of treatments.

02:26 J equals 1 to end the number of treatments.

02:30 And here i have k subj, where k is the number of elements, number of elements in each treatment, each treatment population, times the mean of each treatment minus the grand mean squared.

02:54 So, in this situation, this is equal to 6 times 156 minus 144 squared plus 6 times 142 minus 144 squared plus 6 times 134 minus 144 squared.

03:15 Sorry, that last part's just squeezed in there.

03:19 But this is equal to 14888.

03:24 This is our final value for sum of squares between treatments.

03:29 So i'm going to go ahead and add that to our anova table.

03:33 The sum of squares between treatments is 1488.

03:39 And now we are going to use this value to calculate a mean square between treatments.

03:43 The mean square between treatments is simply the sum of squares between treatments over the number of elements, n minus 1.

03:58 So we calculated the sum of squares between the treatments to be 1488, 1 ,488 over the number of treatments, and we have three treatment populations, a, b, and c.

04:16 So that is 3 minus 1, which is 2.

04:21 So this is equal to 744, 744.

04:28 This is the mean square between our treatment.

04:30 So let me go ahead and update this table.

04:33 So the mean square between our treatments is 744...

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