Question

The following statistics are computed by sampling from three normal populations whose variances are equal: (You may find it useful to reference the t table and the q table.)

xbar = 26.5, n₁ = 7; xbar−2= 32.2, n₂ = 8; xbar 3= 35.7, n₃ = 6; MSE = 33.2

a. Calculate 99% confidence intervals for μ₁ − μ₂, μ₁ − μ₃, and μ₂ − μ₃ to test for mean differences with Fisher’s LSD approach. (Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places. Round your answers to 2 decimal places.)

Population Mean Difference	Confidence Interval	Can we conclude that the population means differ?
mu 1 -mu 2
mu 1- mu 3
mu 2 - mu 3

b. Repeat the analysis with Tukey’s HSD approach. (If the exact value for n_T – c is not found in the table, then round down. Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places. Round your answers to 2 decimal places.)

Population Mean Difference	Confidence Interval	Can we conclude that the population means differ?
mu 1 -mu 2
mu 1- mu 3
mu 2 - mu 3

c. Which of these two approaches would you use to determine whether differences exist between the population means?

• Tukey's HSD Method since it protects against an inflated risk of Type I Error.

• Fisher's LSD Method since it protects against an inflated risk of Type I Error.

• Tukey's HSD Method since it ensures that the means are not correlated.

• Fisher's LSD Method since it ensures that the means are not correlated.

Answer 1

Here we have 3 groups and total number of observations are 7+8+6=21. So degree of freedom is

df= 21-3 = 18

(a)

Critical value of t for and df = 18 using excel function "=TINV(0.01,18)" is 2.878. The Fisher's LSD Value is

The required confidence interval is

groups (i-j)	xbari	xbarj	ni	nj	LSD	xbari-xbarj	absolute diff	Lower limit	Upper limit	Significant(Yes/No)
mu1-mu2	26.5	32.2	7	8	8.58	-5.7	5.7	-14.28	2.88	NO
mu1-mu3	26.5	35.7	7	6	9.23	-9.2	9.2	-18.43	0.03	NO
mu2-mu3	32.2	35.7	8	6	8.96	-3.5	3.5	-12.46	5.46	NO

(b)

Critical value for , df=18 and k=3 is

So Tukey's HSD will be

The required confidence intervals are:

groups (i-j)	xbari	xbarj	ni	nj	HSD	xbari-xbarj	Lower limit	Upper limit	Significant(Yes/No)
mu1-mu2	26.5	32.2	7	8	9.91	-5.7	-15.61	4.21	No
mu1-mu3	26.5	35.7	7	6	10.65	-9.2	-19.85	1.45	No
mu2-mu3	32.2	35.7	8	6	10.34	-3.5	-13.84	6.84	No

(c)

Tukey's HSD Method since it protects against an inflated risk of Type I Error.