Question

	n	x	s
intermittent claudication	73	6.22	1.62
major asymptomatic disease	105	5.81	1.43
minor asymptomatic disease	240	5.77	1.24
no disease	1080	5.47	1.31

One of the goals of the Edinburgh Artery Study was to investigate the risk factors for peripheral arterial disease among persons 55 to 74 years of age. You wish to compare mean LDL cholesterol levels, measured in mmol/liter, among four different populations of subjects: patients with intermittent claudication or interruptions in movement, those with major asymptomatic disease, those with minor asymptomatic disease, and those with no evidence of disease at all. Samples are selected from each population; summary statistics are shown below.

a) test the null hypothesis that the mean LDL cholesterol levels are the same for each of the four populations. What are the degrees of freedom associated with this test?

b) What do you conclude?

c) What assumptions about the data must be true for you to use the one-way analysis of variance technique?

d) Is it necessary to take an additional step in this analysis? If so, what is it? Explain.

Answer 1

Solution:-

The given data represents that the LDL cholestterol levels among four different sub groups.From the given information,

A)

Test the claim that the mean LDL cholesterol levels are the same for each of the four populations.Under the claim the null and alternative hypothesis are,

At least one of the mean is differ from other.

Let us assume the level of significance is

To test the hypothesis,the test statistic is given by,

Compute the estimate of the within groups variance.

Compute the grand mean of the 4 obserfvations that make up the 4 different samples.

Compute the estimate of the between groups variance.

Substitute tghe baove all values in the F statistic:

the degree of freesoms are and

B)

From the F-table values,at 0.05 level of significance and 3,1494 degrees of freedom,the critical value is

Since,test statistic falls in the critical region,we reject the null hypothesis and conclude that the mean LDL cholesterol level of four groups are not identical.

C)

The results of a one-way ANOVA can be considered reliable as long as the following assumption are met:

Responce variable residuals are normally distributed.

Variance of populatons are equal.

Responce for a given group are independent and identically distributed normal random variables.

D)

From the conclusion in Part(A),it is observed that the means are not equal.So,it is necessary to perform an additional step in this analysis.Here,we want to know that which pair means are differ from each other.For this use Bonferroni multiple comparison method.

if we paln to conduct all possible pairwise test and we wish to set the overall probability of making type I error at 0.05,the significance level for an individual comaprison is

Here k is the number of groups.