The U.S. Department of Housing and Urban Development provides data that show the fair market mon

The U.S. Department of Housing and Urban Development...

Key Concepts

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more groups to determine if at least one group mean is statistically significantly different from the others. It achieves this by analyzing the variation between group means relative to the variation within the groups.

Significance Level

The significance level is the probability threshold (commonly set at 0.05) at which the null hypothesis will be rejected. It represents the acceptable risk of committing a Type I error, which is the mistake of rejecting the null hypothesis when it is actually true.

F-Test

The F-test is the test statistic used in ANOVA. It compares the ratio of the between-group variance to the within-group variance. A larger F-value suggests a greater likelihood that the group means are not all equal, leading to the rejection of the null hypothesis if it exceeds a critical value at the chosen significance level.

Null Hypothesis

The null hypothesis represents the idea that there is no difference among the group means. In the context of this test, it means that the true mean fair market monthly rent is equal for all metropolitan areas. This concept serves as the baseline statement to be tested against the observed data.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis, stating that at least one group mean is different from the others. This indicates that not all metropolitan areas have the same mean fair market monthly rent, and it motivates the need for statistical testing in the study.

Transcript

00:01 So in this question, we are trying to see if the means for each geographic region are equal.

00:07 So in order to do that, let's first set up our hypotheses.

00:10 So our null hypothesis would be that the means for each individual region are equal.

00:17 So we have midwest, northeast, south, and west.

00:21 So our mu for the midwest would equal the mu for the northeast, would equal our mu for the south.

00:31 Would equal our mu for the west.

00:36 There's no deviation between our means, is basically what the null is trying to say.

00:41 And our alternative is that not all the means are equal.

00:47 Not all means are equal.

00:54 So this is what we're trying to prove.

00:55 In order to prove this, we're going to go through an annova, and the basic premise of anova is a anova table.

01:03 So what we have to do first is calculate the sum of squares of our treatment.

01:08 In order to calculate the sum of squares of our treatment, we need to come up with the sample means for each of our populations.

01:17 So i set up a table here just to visualize this data more easily.

01:21 So our x bar for the midwest would be 12 .08.

01:26 And i'm going to write out as many decimals as excel goes to in this case.

01:34 So 12 .08125.

01:36 And our x bar for the northeast would be 8 .3625.

01:44 Our x bar for the south would be 12 .016.

01:49 And for the west, it would be 6 .9h repeating.

01:54 And now we have to calculate our sample sizes for each treatment group.

01:59 And notice that the sample sizes are different for each treatment group.

02:02 So in the first treatment, in the midwest sample, we have 16.

02:06 In the northeast sample, we have 16.

02:08 In the south sample, we have 25.

02:11 And in the west sample, we have 18.

02:14 So now using this information, we can come up with a sum of squares for our treatment.

02:19 Oh, almost.

02:20 We also need to come up with a grand mean.

02:23 Then the grand mean is just the average of these means, which is equal to 9 .86216.

02:34 And now for the sum of squares of our treatment, we are going to take the sum of the number of elements in each treatment times the difference between each sample treatment and the grand mean squared.

02:50 So i'll write this down.

02:53 So the first sample midwest has 16 elements, and the sample mean is 12 .08125 minus our grand mean of 9 .86216 squared, plus 16, i'll write this on another line, plus 16 times our second sample mean 8 .3625 minus our grand mean 9 .86216 squared plus 25 times 8 .016 minus 9 .86216 squared plus 15 times 6 .016 squared plus 18 times 6 .016 squared plus 18 times 6 .9 .6216 squared plus 18 times 6 .6.

03:40 9h repeating minus 9 .86216 squared.

03:45 And with this, we get some squares of the treatment to be 379 .3515.

03:54 So now we can update the first value in our anova table.

03:58 We have a value of 379 .3515.

04:07 Now, in your answer, you might not need to be this specific.

04:10 It's just easier because excel gives me this many values.

04:13 So now we have to find the sum of squares for our total, sorry.

04:18 And in order to do that, we are going to take each individual value minus the grand mean squared and sum those squared differences.

04:27 So for example, i will take, so our first value in our data set is 16 .2...