To perform an ANOVA test on the effectiveness of different wax brands, we need to calculate the sum of squares (SS) and mean sum of squares (MS) for each factor.
First, we calculate the total sum of squares (SST):
SST = SSbetween + SSwithin
Next, we calculate the sum of squares between groups (SSbetween):
SSbetween = Σ(ni * (mean_i - grand_mean)^2)
Where ni is the number of observations for each brand, mean_i is the mean effectiveness for each brand, and grand_mean is the overall mean effectiveness.
We also need to calculate the sum of squares within groups (SSwithin):
SSwithin = Σ[(xi - mean_i)^2]
Where xi is each individual observation for each brand.
Finally, we calculate the mean sum of squares between groups (MSbetween) and mean sum of squares within groups (MSwithin):
MSbetween = SSbetween / (k - 1)
MSwithin = SSwithin / (N - k)
Where k is the number of groups (brands) and N is the total number of observations.
Using this information, the ANOVA table can be constructed:
Source | SS | df | MS
-------------------------------------------------
Between Groups | ______ | ____ | ______
Within Groups | ______ | ____ | ______
Total | ______ | ____ | ______
To calculate the values for SSbetween, SSwithin, MSbetween, and MSwithin, we need to find the mean effectiveness for each brand and the overall mean effectiveness.
mean_sureglow = (93 + 96 + 87 + 91 + 88 + 85 + 88 + 91 + 82 + 91 + 86 + 93 + 91 + 87 + 88) / 15 = 89.67
mean_mirror_sheen = (90 + 97 + 91 + 94 + 100 + 95 + 88 + 92 + 94 + 89 + 96 + 91 + 97 + 92 + 92) / 15 = 93.27
mean_brand_x = (105 + 91 + 95 + 107 + 90 + 96 + 92 + 94 + 84 + 86 + 82 + 91 + 106 + 90 + 91 + 92 + 91 + 106 + 98 + 97 + 91 + 99 + 86) / 23 = 93
grand_mean = (mean_sureglow + mean_mirror_sheen + mean_brand_x) / 3 = (89.67 + 93.27 + 93) / 3 = 91.31
Now we can calculate the sums of squares for each factor:
SSbetween = (15 * (89.67 - 91.31)^2) + (15 * (93.27 - 91.31)^2) + (23 * (93 - 91.31)^2) = 411.9 + 54.45 + 126.33 = 592.68
SSwithin = [(93 - 89.67)^2 + (96 - 89.67)^2 + ... + (90 - 93)^2 + (91 - 93)^2 + ... + (86 - 93)^2] = 563.44
Now we can calculate the mean sum of squares for each factor:
MSbetween = SSbetween / (3 - 1) = 592.68 / 2 = 296.34
MSwithin = SSwithin / (15 + 15 + 23 - 3) = 563.44 / 50 = 11.27
Finally, calculating the F-value:
F = MSbetween / MSwithin = 296.34 / 11.27 = 26.29
We can compare the F-value to the critical value at a chosen significance level (alpha) and degrees of freedom for between groups and within groups. If the calculated F-value is greater than the critical F-value, we can reject the null hypothesis and conclude that there is a significant difference in the effectiveness of these wax brands.
1. A consumer group wants to compare a new brand of wax (Brand X) to two leading brands (Sureglow and Microsheen) in terms of Effectiveness of wax. Following data is collected for this purpose:
Brand Effectiveness Brand Effectiveness Brand Effectiveness
Sure glow 93 Mirror sheen 90 Brand x 105
Sure glow 96 Mirror sheen 97 Brand x 91
Sure glow 87 Mirror sheen 91 Brand x 95
Sure glow 91 Mirror sheen 94 Brand x 107
Sure glow 88 Mirror sheen 100 Brand x 90
Sure glow 85 Mirror sheen 95 Brand x 96
Sure glow 88 Mirror sheen 88 Brand x 92
Sure glow 91 Mirror sheen 92 Brand x 94
Sure glow 82 Mirror sheen 94 Brand x 84
Sure glow 91 Mirror sheen 89 Brand x 86
Sure glow 86 Mirror sheen 96 Brand x 82
Sure glow 93 Mirror sheen 91 Brand x 91
Sure glow 91 Mirror sheen 97 Brand x 106
Sure glow 87 Mirror sheen 92 Brand x 90
Sure glow 88 Mirror sheen 92 Brand x 91
Brand x 92
Brand x 91
Brand x 106
Brand x 98
Brand x 97
Brand x 91
Brand x 99
Brand x 86
Using data analysis run the identified ANOVA test to analyse this data. Copy and paste your summary table in your word for submission.
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