Question

Suppose the salinity of water samples taken from 3 new and separate sites in the Bimini Lagoon was measured with the results given below:

Site 1: 49.59 49.60 54.11 50.06 47.42 45.40 42.32
Site 2: 54.66 54.67 59.74 55.19 52.22 49.95 46.50 54.69 54.95 45.04 54.88 54.02
Site 3: 45.65 45.66 50.72 46.18 43.22 40.95 37.50 45.68 45.94 36.05 45.87

Basic Statistics for the 3 samples:

Level	N	Mean	SD
1	7	48.36	3.77
2	12	53.04	4.07
3	11	43.95	4.25

One-way ANOVA output:

Source	DF	SS	MS	F	P
Factor	2	474.81	237.41	14.30	0.0001
Error	27	448.12	16.60
Total	29	922.93

Using the data above, construct simultaneous (1−α*)100% confidence intervals for the pairwise differences in means using the Bonferroni procedure. Use an overall experiment-wide α = 0.05.

1. What is the value of α^*? Use at least 4 digits to the right of the decimal.

2. The critical t-value has a student's t distribution with how many degress of freedom?

3. What is the critcal t-value? You can use the invcdf command in Minitab or the qt() function in R.

4.Calculate the confidence interval for the difference in means for Levels 1 and 2, i.e., μ₁−μ₂. Lower bound: Upper bound:

5. What is your conclusion about the relative size of the means for Levels 1 and 2?

6. Calculate the confidence interval for the difference in means for Levels 1 and 3, i.e., μ₁−μ₃. Lower bound:  Upper bound:

7. What is your conclusion about the relative size of the means for Levels 1 and 3?

8. Calculate the confidence interval for the difference in means for Levels 2 and 3, i.e., μ₂−μ₃. Lower bound:  Upper bound:

9. What is your conclusion about the relative size of the means for Levels 2 and 3?

Answer 1

1) The Bonferroni correction sets the significance cut-off at α/n = 0.05/3 = 0.0167

2) DF error =N-k=   27

3) t-critical value,t(α/2,df)=   2.5525

4)

bonferroni critical value=tα/2,df √(MSE(1/ni+1/nj))
confidence interval = mean difference ± critical value          
          

			confidence interval
population mean difference		critical value	lower limit	upper limit
µ1-µ2	-4.68	4.95	-9.6260	0.2660

5) µ1 = µ2

6)

			confidence interval
population mean difference		critical value	lower limit	upper limit
µ1-µ3	4.41	5.03	-0.6181	9.4381

7) conclusion: µ1 = µ3

8)

			confidence interval
population mean difference		critical value	lower limit	upper limit
µ2-µ3	9.09	4.34	4.7490	13.4310

9) conclusion: µ2 > µ3