4.5.2 Inferential Statistics Levene's test an equal variance test-was initially employed to determine if our data sets fulfil the homogeneity of variance assumption before performing Analysis of Variance (ANOVA). The null hypothesis (Ho) and alternative hypothesis (H) are as follows: • Ho: Concentration groups have equal variances. H₁: Concentration groups have different variances. The p-values for both S. aureus and E. coli data sets are less that 0.05, indicating there is a significant difference between the variances, thus null hypothesis is rejected (Table 4.8). Since the homogeneity assumption of the variance is not violated, non-parametric test like Kruskal-Wallis, which is an alternative to one-way ANOVA, test should be used. Table 4.8: Test for equality of variances (Levene's) Bacteria F dfl df2 S. aureus 9.600 5.000 12.000 Р <.001 E. coli 6.000 5.000 12.000 < .001 Next, Kruskal-Wallis test determines whether the medians of two or more groups are different, and the hypothesis are as follows: • Ho: There is no significant difference in the median of the concentration groups. H₁: At least one concentration group has a different median compared to the other groups. If the p-value is larger than 0.05, the null hypothesis is retained, otherwise it is rejected. In this case, the p-values for both S. aureus and E. coli are 0.009 and 0.006, respectively, which are less than 0.05 (Table 4.9). The null hypothesis is thus rejected, and it is assumed that there is a significant difference between the different groups relative to its concentrations. Bacteria Factor S. aureus Concentration E. coli Concentration Table 4.9: Kruskal-Wallis test Statistic df P 15.264 5 0.009 16.399 5 0.006 The Kruskal-Wallis test does not provide an answer to the question of which of the concentration groups differed for both cultures; thus Dunn's post-hoc test is required to further address this issue for the pairwise multiple comparison. As usual, if the p-value in a pairwise comparison is less than the significance level (i.e., 0.05), the null hypothesis that there is no difference is thus rejected, and it is assumed that the respective two groups differ. For S. aureus, there is a significant difference when increasing the concentration from 50 mg/ml to 200 mg/ml, 250 mg/ml, 300 mg/ml or 400 mg/ml as well as from 100 mg/ml to 400 mg/ml (Table 4.10). On the other hand, for E. coli, increasing the concentration from 50 mg/ml to 250 mg/ml, 300 mg/ml or 400 mg/ml as well as from 100 mg/ml to 300 mg/ml or 400 mg/ml mark a significant difference (Table 4.11). Note that the p-value is unadjusted, acknowledging the increased risk of Type I errors. However, if Bonferroni adjustment is applied, the significant comparison is only seen when increasing the concentration from 50 mg/ml to 400 mg/ml for both cultures (Table 4.10 and 4.11). Table 4.10: Dunn's post hoc comparisons for S. aureus - concentration Comparison Ꮓ Wi Wj р Pbonf 50 mg/ml 100 mg/ml -1.066 2.000 6.333 0.286 1.000 50 mg/ml - 200 mg/ml -1.969 2.000 10.000 0.049 * 0.735 50 mg/ml - 250 mg/ml -1.969 2.000 10.000 0.049 * 0.735 50 mg/ml - 300 mg/ml -2.420 2.000 50 mg/ml -400 mg/ml -3.651 2.000 100 mg/ml 200 mg/ml 100 mg/ml 250 mg/ml 100 mg/ml 300 mg/ml -0.902 6.333 -0.902 6.333 -1.354 6.333 16.833 10.000 0.367 11.833 0.016 <.001 * 0.233 *** 0.004 ** 1.000 10.000 0.367 1.000 11.833 0.176 1,000

4.5.2 Inferential Statistics Levene's test an equal variance test-was initially employed to determine if our data sets fulfil the homogeneity of variance assumption before performing Analysis of Variance (ANOVA). The null hypothesis (Ho) and alternative hypothesis (H) are as follows: • Ho: Concentration groups have equal variances. H₁: Concentration groups have different variances. The p-values for both S. aureus and E. coli data sets are less that 0.05, indicating there is a significant difference between the variances, thus null hypothesis is rejected (Table 4.8). Since the homogeneity assumption of the variance is not violated, non-parametric test like Kruskal-Wallis, which is an alternative to one-way ANOVA, test should be used. Table 4.8: Test for equality of variances (Levene's) Bacteria F dfl df2 S. aureus 9.600 5.000 12.000 Р <.001 E. coli 6.000 5.000 12.000 < .001 Next, Kruskal-Wallis test determines whether the medians of two or more groups are different, and the hypothesis are as follows: • Ho: There is no significant difference in the median of the concentration groups. H₁: At least one concentration group has a different median compared to the other groups. If the p-value is larger than 0.05, the null hypothesis is retained, otherwise it is rejected. In this case, the p-values for both S. aureus and E. coli are 0.009 and 0.006, respectively, which are less than 0.05 (Table 4.9). The null hypothesis is thus rejected, and it is assumed that there is a significant difference between the different groups relative to its concentrations. Bacteria Factor S. aureus Concentration E. coli Concentration Table 4.9: Kruskal-Wallis test Statistic df P 15.264 5 0.009 16.399 5 0.006 The Kruskal-Wallis test does not provide an answer to the question of which of the concentration groups differed for both cultures; thus Dunn's post-hoc test is required to further address this issue for the pairwise multiple comparison. As usual, if the p-value in a pairwise comparison is less than the significance level (i.e., 0.05), the null hypothesis that there is no difference is thus rejected, and it is assumed that the respective two groups differ. For S. aureus, there is a significant difference when increasing the concentration from 50 mg/ml to 200 mg/ml, 250 mg/ml, 300 mg/ml or 400 mg/ml as well as from 100 mg/ml to 400 mg/ml (Table 4.10). On the other hand, for E. coli, increasing the concentration from 50 mg/ml to 250 mg/ml, 300 mg/ml or 400 mg/ml as well as from 100 mg/ml to 300 mg/ml or 400 mg/ml mark a significant difference (Table 4.11). Note that the p-value is unadjusted, acknowledging the increased risk of Type I errors. However, if Bonferroni adjustment is applied, the significant comparison is only seen when increasing the concentration from 50 mg/ml to 400 mg/ml for both cultures (Table 4.10 and 4.11). Table 4.10: Dunn's post hoc comparisons for S. aureus - concentration Comparison Ꮓ Wi Wj р Pbonf 50 mg/ml 100 mg/ml -1.066 2.000 6.333 0.286 1.000 50 mg/ml - 200 mg/ml -1.969 2.000 10.000 0.049 * 0.735 50 mg/ml - 250 mg/ml -1.969 2.000 10.000 0.049 * 0.735 50 mg/ml - 300 mg/ml -2.420 2.000 50 mg/ml -400 mg/ml -3.651 2.000 100 mg/ml 200 mg/ml 100 mg/ml 250 mg/ml 100 mg/ml 300 mg/ml -0.902 6.333 -0.902 6.333 -1.354 6.333 16.833 10.000 0.367 11.833 0.016 <.001 * 0.233 * 0.004 1.000 10.000 0.367 1.000 11.833 0.176 1,000

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.4: Distributions Of Data

Problem 19PFA

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