If the null hypothesis is rejected in a one-way ANOVA test...

If the null hypothesis is rejected in a one-way ANOVA test of three or more means, then a Scheffé Test can be performed to find which means have a significant difference. In a Scheffé Test, the means are compared two at a time. For instance, with three means you would have the following comparisons: $\bar{x}_{1}$ versus $\bar{x}_{2}, \bar{x}_{1}$ versus $\bar{x}_{3},$ and $\bar{x}_{2}$ versus $\bar{x}_{3} .$ For each comparison, calculate $$\frac{\left(\bar{x}_{a}-\bar{x}_{b}\right)^{2}}{\frac{S S_{W}}{\Sigma\left(n_{i}-1\right)}\left(\frac{1}{n_{a}}+\frac{1}{n_{b}}\right)}$$ where $\bar{x}_{a}$ and $\bar{x}_{b}$ are the means being compared and $n_{a}$ and $n_{b}$ are the corresponding sample sizes. Calculate the critical value by multiplying the critical value of the one-way ANOVA test by $k-1 .$ Then compare the value that is calculated using the formula above with the critical value. The means have a significant difference when the value calculated using the formula above is greater than the critical value. Refer to the data in Exercise $8 .$ At $\alpha=0.01,$ perform a Scheffé Test to determine which means have a significant difference.

Key Concepts

Critical Value and Significance Level

The critical value is a threshold used to decide whether an observed test statistic is large enough to reject the null hypothesis, based on the chosen significance level (alpha). In the Scheffé test, the critical value is adjusted by the factor (k-1), where k is the number of groups, to account for the increased chance of Type I error from multiple comparisons.

Scheffé Test

The Scheffé test is a post hoc multiple comparison procedure that is applied after a significant ANOVA result. It is designed to compare all possible pairs (or contrasts) of group means and is known for its conservative nature, meaning it controls the overall Type I error rate effectively when performing numerous comparisons.

Multiple Comparisons Correction

Multiple comparisons correction refers to the statistical adjustments made when conducting several tests simultaneously, such as comparing every pair of group means, to reduce the likelihood of Type I errors. The Scheffé test accomplishes this by adjusting its critical value according to the number of groups, ensuring that the probability of falsely declaring a significant difference is maintained at the desired level.

One-Way ANOVA

One-way analysis of variance (ANOVA) is a statistical method used to test the hypothesis that several group means are equal. It decomposes the total variation observed in the data into between-group variation and within-group variation, and it uses the F-statistic to determine if the observed differences among means could be due to random chance.

Null Hypothesis

In the context of ANOVA, the null hypothesis states that all group means are equal, implying that any differences observed between sample means are due only to sampling variability. Rejection of this hypothesis indicates that at least one group mean is statistically different from the others.

Transcript

00:01 The following is a solution video to number 21 using the chaffaise test and determining which mean of these three categories is significantly different from the other.

00:12 And we have, these are mean or median salaries or something and there are federal employees, state employees, and local employees for different governments.

00:22 Okay, so here's your data set.

00:24 And the first thing that i found was the average for federal and state and then local.

00:30 So the way you do that is equal average a2 to a11, and then that's for federal.

00:36 So the federal average is about 73 ,000, and then 50 ,000 for the state, and then about 44 ,000 for the local.

00:45 So those are the averages.

00:46 Okay, the second thing that i found was the sample variance, and there may be an easier way to do this, but i just did the standard deviation, the sample standard deviation, so make sure it's dot s, not dot p, because it's a sample, not a population.

00:59 And of the data set.

01:01 And then i squared it.

01:02 So remember that the sample variance is just the sample standard deviation squared.

01:07 So the sample variance for federal is 93 .75.

01:12 For the state, it's about 57.

01:15 And then for the local, it's about 62 .4.

01:18 So those are the sample variances, because that plays a part in the formula as well.

01:23 And then we have these n minus 1.

01:24 So that's the sample size minus 1, because that's the other part of the formula.

01:28 So each of these sample sizes are 10.

01:30 So 10 minus 1 is 9.

01:32 And then i have this ssw down here.

01:34 So i went ahead and solved that.

01:37 But it is, and this is in your book, but it's the n minus 1, it's the summation of n minus 1 times the sample variance.

01:44 So that's why i did this a20 times a17 plus b20 times b17 plus c20 times c17.

01:52 And that gave me 1918 .441.

01:55 So that's the ssw.

01:57 And then the other part of that formula is the summation of the n minus 1.

02:01 So that's just this row here added together.

02:05 So 9 plus 9 plus 9 is 27.

02:07 Okay, so over here i have the f star, and that's the critical value.

02:10 And the way the book says to find the critical value is you just take the original critical value that we found back in the original example with this, and it was 5 .49.

02:21 And you can look back at, you know, we already did this problem, so 5 .49, and then times the number of categories minus 1.

02:28 So the number of categories, there are three categories, federal, state, and local, minus 1 is 3.

02:33 So i took that 5 .49 times 2.

02:36 So my critical value is 10 .98.

02:38 So the rule here is if our sheffay statistic is larger than 10 .98, then there is a significant difference between those means.

02:47 If it's less than 10 .98, then there's not a significant difference between the means...

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