Question

4 A hospital wants to determine if there is a difference in time to recover from an illness for treatment with three different treatments. Sample data is as follows: Antibiotic A: Sample mean4 dayss-1.4 days -2 days n 10 days s-1.4 days -2 days n-10 x=6days s-1.4 days s2 = 2 days n=10 Antibiotic B: Sample mean 5 Antibiotic C: Sample mean Grand mean5 days Can you prove at 95% confidence that there is a difference between the three antibiotics? To get full credit, state your null and alternate hypotheses, show how you calculated your test statistic, show your critical region, and state your conclusion: is there a difference between the three antibiotics? YES or NO

Answer 1

We Use a One Way ANOVA for the above data

The Hypothesis:

H0: There is no difference between the means of the time taken to recover from the three treatments

Ha: The mean time of at least 1 of the three treatments is different from the others.

The ANOVA table is as below. the p value is calculated for F = 2.00 for df1 = 2 and df2 = 27

The Fcritical is calculated at $\alpha$ = 0.05 for df1 = 2 and df2 = 27

Source	SS	DF	MS	F	Fcr	p value
Between	20.0	2	10.0	5.0	3.3541	0.0142
Within	54.0	27	2.0
Total	74.0	29

The Decision Rule:

If Ftest is > F critical, Then Reject H0.

Also if p-value is < $\alpha$ , Then reject H0.

The Decision:

Since Ftest (5.0) is > F critical (3.3541), We Reject H0.

Also since p-value (0.0142) is < $\alpha$ (0.05), We reject H0.

The Conclusion: There is sufficient evidence at the 95% level of significance to conclude that the mean time of at least 1 of the three treatments is different from the others.

YES, There is a difference in the 3 antibiotics.

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The Calculations for ANOVA are below:

The overall mean = 5

SS treatment = SUM n * ( $\bar{x_i}$ - overall mean)² = 10 * (4 - 5)² + 10 * (5 - 5)² + 10 * (6 - 5)² = 10 + 0 + 10 = 20

df1 = k - 1 = 3 - 1 = 2

MSTR = SS treatment / df1 = 20 / 2 = 10

SSerror = SUM [(n - 1) * (Variance)] = (9 * 2) + (9 * 2) + (9 * 2) = 54

df2 = N - k = 30 - 3 = 27

Therefore MS error = SSerror / df2 = 54 / 27 = 2

F = MSTR / MSE = 10 / 2 = 5

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