(a) Test for existence of a difference in the doses used- (b) Discuss the suitability of the test used above- (c) Using 10 examples elaborate how you would apply this test in public health-

To test for the existence of a difference in the doses used, we can use a statistical test called Analysis of Variance (ANOVA). ANOVA is suitable for comparing the means of more than two groups, in this case, the four different groups (A, B, C, and D). Here's how you can perform the ANOVA test step by step:

Step 1: Formulate the null hypothesis (H0) and the alternative hypothesis (Ha):

• Null Hypothesis (H0): There is no significant difference in the length of hospital stay among the four groups (A, B, C, D).
• Alternative Hypothesis (Ha): There is a significant difference in the length of hospital stay among the four groups (A, B, C, D).

Step 2: Calculate the mean length of hospital stay for each group:

• Group A: Mean = [(3 * 6) + (4 * 14)] / (6 + 14) = (18 + 56) / 20 = 74 / 20 = 3.7
• Group B: Mean = [(4 * 18) + (5 * 2)] / (18 + 2) = (72 + 10) / 20 = 82 / 20 = 4.1
• Group C: Mean = [(4 * 10) + (5 * 9) + (6 * 1)] / (10 + 9 + 1) = (40 + 45 + 6) / 20 = 91 / 20 = 4.55
• Group D: Mean = [(4 * 8) + (5 * 12)] / (8 + 12) = (32 + 60) / 20 = 92 / 20 = 4.6

Step 3: Calculate the overall mean length of hospital stay (grand mean):

Grand Mean = [(3.7 * 20) + (4.1 * 20) + (4.55 * 20) + (4.6 * 20)] / (20 + 20 + 20 + 20) = (74 + 82 + 91 + 92) / 80 = 339 / 80 = 4.2375

Step 4: Calculate the sum of squares between groups (SSB):

SSB = Σ(group size) * (group mean - grand mean)^2

For Group A: SSB_A = 20 * (3.7 - 4.2375)^2 = 20 * (-0.5375)^2 = 20 * 0.2886 = 5.772

Similarly, calculate SSB for Groups B, C, and D.

SSB_B = 20 * (4.1 - 4.2375)^2 = 20 * (-0.1375)^2 = 20 * 0.0189 = 0.378 SSB_C = 20 * (4.55 - 4.2375)^2 = 20 * 0.0972^2 = 20 * 0.0094 = 0.188 SSB_D = 20 * (4.6 - 4.2375)^2 = 20 * 0.3625^2 = 20 * 0.1313 = 2.626

Step 5: Calculate the sum of squares within groups (SSW):

SSW = Σ(group size - 1) * group variance

For Group A: SSW_A = (6 - 1) * [(3 - 3.7)^2 + (4 - 3.7)^2] = 5 * [(-0.7)^2 + (-0.7)^2] = 5 * [0.49 + 0.49] = 5 * 0.98 = 4.9

Similarly, calculate SSW for Groups B, C, and D.

SSW_B = (18 - 1) * [(4 - 4.1)^2 + (5 - 4.1)^2] = 17 * [0.01 + 0.81] = 17 * 0.82 = 13.94 SSW_C = (10 - 1) * [(4 - 4.55)^2 + (5 - 4.55)^2 + (6 - 4.55)^2] = 9 * [0.3025 + 0.2025 + 1.2025] = 9 * 1.7075 = 15.3675 SSW_D = (12 - 1) * [(4 - 4.6)^2 + (5 - 4.6)^2] = 11 * [0.36 + 0.16] = 11 * 0.52 = 5.72

Step 6: Calculate the degrees of freedom (DF):

• DF between groups (DFB) = Number of groups - 1 = 4 - 1 = 3
• DF within groups (DFW) = Total number of observations - Number of groups = 80 - 4 = 76