Statistics QUESTIONS 2 AND 3

Statistics

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Statistics

QUESTIONS 2 AND 3

Since the probability value of the samples (p-value) = 0.07348>0.05, we fail to reject the null hypothesis that the average salaries by gender are equal and conclude that the samples are not statistically significant.

We also see that the probability value (p-value)=0.17231>0.05=α, and so conclude that there is no significant difference in the interaction between salaries and grades.

QUESTIONS 4 AND 5

Gender A B C D E F

M 24.5 27.5 43.5 48 61 76.5

F 23 35 41.5 53.5 67 76

The following applies for a 2-way ANOVA without replications

We use δij for the effect of level i of factor A with level j of factor B, i.e. the interaction of level i of factor F and level j of factor M. δij = μij – μi – μj + μ.

Finally, we can represent each element in the sample as:

where εijk denotes the error (or unexplained) amount. As before we have the sample version

H0: μ1. = μ2. = … = μr. (Factor F)

H0: μ.2 = μ.2 = … = μ.c (Factor M)

These are equivalent to:

H0: αi = 0 for all i (Factor F)

H0: βj = 0 for all j (Factor M)

In addition there is a null hypothesis for the effects due to interaction between factors A and B.

H0: δij = 0 for all i, j

We can also define the following entities:

Where SS refers to The sum of squaresand df as degrees of freedom.

Assumptions:

All samples are drawn from normally distributed populations

All populations have a common variance

All samples are drawn independently from each other

Within each sample, the observations are sampled randomly and independently of each other

The following tests can be done when it comes to a 2-way ANOVA without replication with variables defined differently. But the principle remains the same;

Reference

Mittelhammer, R. (2013). Mathematical Statistics for Economics and Business. New York: Springer.