SST in Excel ▪ Column 1 = xij - Shows each data value in a vertical column ▪ Column 2 = x x - Shows the same grand mean in every column ▪ Column 3 = (xij – ) x x - (Column 1 – Column 2) ▪ Column 4 = (xij – )² x x - Square the data points in column 3 ▪ ANSWER = sum of data points in column 4 o Mean Square Total (MST) – another term for the variance of the sample data ▪ FORMULA ▪ MST = (SST) ÷ (n – 1) ▪ In our example:
- 15 data points - SST = 833.73 - MST = (833.73) ÷ (15 – 1) = 59.55 o Notes: ▪ If null hypothesis is true (assumed unless proven otherwise) ▪ → levels being compared are from same population with: - Mean = µ - Variance = σ² ▪ THEN MST is a good estimate for the population variance σ² ● Step 4: Partition SST the total sum of squares (SST) into the sum of squares b/w (SSB) and the sum of squares w/in (SSW) o SST = divided into 2 components ▪ The sum of squares b/w SSB ▪ The sum of squares w/in SSW ▪ SST = SSB + SSW o Sum of squares between SSB (SSB) – measures the variation between each sample mean and the grand mean of the data ▪ (sample mean – grand mean)² • (number of data points in sample) ▪ Find this for each sample population ▪ Add up the results for → SSB o Mean Square Between MSB (MSB) – provides an estimate for the population variance (σ²) if the null hypothesis is true ▪ (SSB) ÷ (k – 1) ▪ Where k = number of populations being compared - Ex: if 4 types of smartphones being compared, k = 4 o Sum of Squares within SSW (SSW) – measures the variation between each data value and the corresponding sample mean ▪ SSW = (SST – SSB) o Mean Square within MSW (MSW) – provides another estimate for the population variance (σ²)
▪ (SSW) ÷ (n – k) - n = number of observations (total data points) - k = number of populations being observed (number of sample populations) ● Step 5: Calculate the appropriate test statistic ▪ Use the F-test statistic for one-way ANOVA - Ronald Fisher = statistician o Formula – F-test Statistic for One-Way ANOVA ▪ = (MSB) ÷ (MSW ▪ One way ANOVA equations = summarized on page 519 o IF null hypothesis was true ▪ MSB and MSW would be close to one another ▪ Critical F-score would be close to 1.0 o IF null hypothesis is NOT true ▪ Expect MSB to be larger than MSW ▪ Critical F-score exceeds 1.0 ● Step 6: Determine the appropriate critical value o F-distribution ▪ Righ-skewed ▪ Rejection region in right tail ▪ ALWAYS one-tail (upper) hypothesis test ▪ Critical value (F∞) = found in Appendix A o 2 types of Degrees of Freedom ▪ D1 & D2 ▪ Correspond to degrees of freedom for: - Sum of squares b/w SSB - Sum of squares w/in SSW ▪ D1 (COLUMN) = k – 1 - Ex: (4 – 1) = 3 ▪ D2 (ROW) = n – k - Ex: (15 – 4) = 11
▪ Column 3 / Row 11, where ∞ = 0.05 - Critical value of 3.587 ● Step 7: compare the F-test statistic (F) with the critical (F-score) o Decision Rules – ANOVA ▪ F ≤ F-score → DO NOT REJECT H0 ▪ F > F-score → REJECT H0 o In our example: ▪ F-score = 3.587 and F = 9.86 ▪ F > F-score → REJECT ● Step 8: State your conclusions o Reject the null hypothesis ▪ Conclude that not all 4 population means are equal (some difference) ▪ NOT enough info to decide which populations have significantly different averages (find this out later) o ANOVA = compares 2 types of variances ▪
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- Normal Distribution, Null hypothesis, Hypothesis testing, Statistical hypothesis testing