In the City of Cleveland appraisals for taxes purposes is conducted every 4 years. The county assessor believed there was variance in the different people that were sent to perform the assessments. A test was developed to determine if there was difference in the assessors. Use a 0.05 alpha to determine if there is any difference in the assessments among the assessors and is there any difference among the houses related to the assessment value (expressed in thousands of dollars)
ASSESSORS
House Smith Jones Wylie Myers
A 53.0 55.0 49.0 34.0
B 50.0 51.0 52.0 53.0
C 48.0 52.0 47.0 53.0
D 70.0 68.0 65.0 64.0
E 84.0 89.0 92.0 86.0
To determine if there is any difference in the assessments among the assessors, we can use an analysis of variance (ANOVA) test. ANOVA compares the means of different groups to determine if there is a significant difference between them.
Here's how you can perform the ANOVA test using a significance level (alpha) of 0.05:
Step 1: State the null hypothesis and alternative hypothesis:
- Null hypothesis (H0): There is no difference in the assessments among the assessors.
- Alternative hypothesis (HA): There is a difference in the assessments among the assessors.
Step 2: Calculate the sum of squares between groups (SSB) and the sum of squares within groups (SSW):
- SSB measures the variation between the different assessors' assessments.
- SSW measures the variation within each assessor's assessments.
Step 3: Calculate the mean square between groups (MSB) and the mean square within groups (MSW):
- MSB is obtained by dividing the SSB by the degrees of freedom between groups (dfB).
- MSW is obtained by dividing the SSW by the degrees of freedom within groups (dfW).
Step 4: Calculate the F-statistic:
- The F-statistic is calculated by dividing MSB by MSW.
Step 5: Determine the critical F-value:
- Look up the critical F-value for the given significance level (alpha) and the degrees of freedom between groups (dfB) and within groups (dfW) using an F-table.
Step 6: Compare the F-statistic with the critical F-value:
- If the F-statistic is greater than the critical F-value, reject the null hypothesis and conclude that there is a significant difference in the assessments among the assessors.
- If the F-statistic is less than or equal to the critical F-value, fail to reject the null hypothesis and conclude that there is no significant difference in the assessments among the assessors.
To perform the ANOVA test, let's calculate the necessary values:
First, calculate the sum of squares total (SST), which is the sum of the squared differences between each observation and the grand mean:
SST = Σ(xi - x̄)^2
where xi is each observation and x̄ is the grand mean.
Next, calculate the sum of squares between groups (SSB), which is the sum of the squared differences between each group mean and the grand mean, weighted by the number of observations in each group:
SSB = Σ(ni * (x̄i - x̄)^2)
where ni is the number of observations in each group, x̄i is the mean of each group, and x̄ is the grand mean.
Then, calculate the sum of squares within groups (SSW), which is the sum of the squared differences between each observation and its respective group mean:
SSW = Σ(xi - x̄i)^2
where xi is each observation in a specific group and x̄i is the mean of each group.
Finally, calculate the degrees of freedom between groups (dfB) and within groups (dfW):
- dfB = number of groups - 1
- dfW = total number of observations - number of groups
Now that we have all the necessary values, we can proceed to calculate the mean squares and the F-statistic. After that, we can compare the F-statistic with the critical F-value to make a conclusion.