Question

Consider the following measurements of blood hemoglobin concentrations (in g/dL) from three human populations at different geographic locations:

population1 = [ 14.7 , 15.22, 15.28, 16.58, 15.10 ]

population2 = [ 15.66, 15.91, 14.41, 14.73, 15.09]

population3 = [ 17.12, 16.42, 16.43, 17.33]

Perform ANOVA to check if any of these populations have different mean hemoglobin concentrations. (Assume that all the ANOVA requirements such as normality, equal variances and random samples are met.) After you perform ANOVA perform a Tukey-Kramer post-hoc test at a significance level of 0.05 to see which populations actually have different means. As usual, round all answers to two digits after the decimal point. (Make sure you round off to at least three digits any intermediate results in order to obtain the required precision of the final answers.) For any questions, which ask about differences in means or test statistics, which depend on differences in means provide absolute values. In other words if you get a negative value, multiply by -1 to make it positive.

QUESTION 1

1. For the three populations, what is the value of SSgroups in the ANOVA table?

QUESTION 2

1. For the three populations, what is the value of SSerror in the ANOVA table?

QUESTION 3

1. For the three populations, how many degrees of freedom are there for the groups?  

QUESTION 4

1. For the three populations, how many degrees of freedom are there for the error?

Answer 1

Here we have performed One-way ANOVA with Tukey's Post-hoc test al 0.05 level.

MINITAB gives the following result

SSgroups= 7.043 df for gropus=2

SSerror= 4.243 df for error=11

MINITAB output:

One-way ANOVA: Hemoglobin versus Population

Source DF SS MS F P
Group 2 7.043 3.522 9.13 0.005
Error 11 4.243 0.386
Total 13 11.286

S = 0.6211 R-Sq = 62.40% R-Sq(adj) = 55.57%

Since p-value<0.05, reject the null hypothsis and at 5% level we can conclude that the mean hemoglobin level for three groups are NOT equal.

Individual 95% CIs For Mean Based on
Pooled StDev
Level N Mean StDev --------+---------+---------+---------+-
1 5 15.376 0.710 (------*-------)
2 5 15.160 0.625 (------*-------)
3 4 16.825 0.470 (-------*--------)
--------+---------+---------+---------+-
15.20 16.00 16.80 17.60

Pooled StDev = 0.621

Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of Population

Individual confidence level = 97.94%

Population = 1 subtracted from:

Population Lower Center Upper ---------+---------+---------+---------+
2 -1.2770 -0.2160 0.8450 (-------*------)
3 0.3236 1.4490 2.5744 (-------*------)
---------+---------+---------+---------+
-1.5 0.0 1.5 3.0

Population = 2 subtracted from:

Population Lower Center Upper ---------+---------+---------+---------+
3 0.5396 1.6650 2.7904 (------*-------)
---------+---------+---------+---------+
-1.5 0.0 1.5 3.0

Since the CI for pairwise difference in means for Population 1 and 2 includes '0', we can conclude that the difference of mean hemoglobin level for pair (group 1, group 3) and (group 2, group 3) are dignificantly different whereas it is insignificant for pair (group 1, group 2).