Develop the analysis of variance computations for the following completely randomized design. At

Develop the analysis of variance computations for the...

Key Concepts

Analysis of Variance (ANOVA)

ANOVA is a statistical technique used to compare the means of multiple groups to determine if at least one group mean significantly differs from the others. It partitions the total variability observed in the data into components attributed to differences between the groups (treatments) and within the groups (error), allowing for a systematic evaluation of treatment effects.

Completely Randomized Design

A completely randomized design is an experimental setup in which all experimental units are assigned to treatments purely by chance. This randomization helps to mitigate the effects of confounding variables, ensuring that treatment differences can be more reliably attributed to the treatments themselves rather than other external factors.

Hypothesis Testing in ANOVA

The fundamental framework of hypothesis testing in ANOVA involves setting up a null hypothesis that states all treatment means are equal, and an alternative hypothesis that at least one treatment mean differs. This framework provides a clear criterion for assessing whether the observed differences in data are statistically significant or could have occurred by random chance.

Sum of Squares and Mean Squares

In the context of ANOVA, the total variability in data is quantified by the Total Sum of Squares (SST), which is then split into the Treatment Sum of Squares (SSB) representing variability between treatment means, and the Error Sum of Squares (SSE) representing variability within treatments. Mean squares are obtained by dividing these sums of squares by their corresponding degrees of freedom, providing estimates of variance that are crucial for the subsequent F-test.

F-test

The F-test in ANOVA is used to compare the variance estimates between treatments and within treatments. By computing the ratio of the Mean Square Between Treatments (MSB) to the Mean Square Error (MSE), the F-test assesses whether the treatment effect is significantly greater than the variability from random error. This F-statistic is compared to a critical value from the F-distribution to determine statistical significance.

Degrees of Freedom and Significance Level

Degrees of freedom in ANOVA are linked with the number of groups and the total sample size, with different df values for the treatment and error components. These values are key in computing the mean squares and determining the correct F-distribution for testing. The significance level (often denoted by alpha, such as .05) defines the threshold for rejecting the null hypothesis, indicating the probability of making a Type I error.

Transcript

00:01 We will be assuming that those three treatment groups have equal means, and alternately that not all means, population means, are equal.

00:18 And i believe your book uses that h -sub -a.

00:22 And so we'll be filling in an anewa table, but let's go through and write down the information we know.

00:28 We know these x -s -bars.

00:30 We know 119, 107, and 100.

00:35 And then we know what the variances are for each of these groups.

00:39 And this is 146 .86.

00:42 This is 96 .44 and 173 .78.

00:48 So we'll be filling in this table with we'll have a treatment group, the air, we'll have the total, and we'll have the sum of squares, we'll have the degrees of freedom, and we'll have the, let's see, the mean square, the f statistic, and then the p value.

01:10 So we basically will fill out all these categories.

01:14 And so the first thing we need to do is, well, let's do the degrees of freedom.

01:17 Our degrees of freedom, and let me put the sample size we have for each of these.

01:21 This has a sample size of eight, this of ten, and this of ten.

01:25 So that grand total is ten, twenty, eight.

01:30 And so our degrees of freedom, we have three categories.

01:34 So this will have k minus two or three minus one is two.

01:39 Our air is going to end up being the 28, the 28 minus the number of categories of just three.

01:47 So this will be 25.

01:49 And so we need to find these totals.

01:52 So let's find the sum of squares for the treatment, first of all.

01:56 And we know in order to do that, we need to take the sample.

01:59 Size of each times the difference between that and the grand mean.

02:05 And our grand mean is going to be calculated by taking eight times this 146 .86 plus 10 times that 96 .44 plus 10 times i just looked at the wrong number.

02:26 This is 119.

02:28 Let me clear that out.

02:29 I'm looking at the wrong numbers.

02:31 We have eight times 119 plus 10 times 10 plus 100 times 10.

02:42 And then we'll divide that by that total of 28.

02:45 So i end up having 3 ,022 divided by 28.

02:50 And that gives me a mean of 107 .9.

02:55 I'm going to round it to three.

02:57 And i'm actually going to actually store it in my calculator as x.

03:03 So then when i'm finding this mean for that treatment, and i don't want to just add these three together and divide because we want what the actual brand mean is of those three.

03:21 So now we'll go through and we'll find that some squares for the, some of the squares for the treatment...

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