## Importance of the Normal Distribution

### Parametric Hypothesis Testing

All parametric hypothesis testing that we're going to perform requires normality in some sense.

Population Mean
Either the population was normally distributed, the sample size was large enough (so the central limit theorem applied and was approximately normal), or the population was approximately normal and the student's t was used.
Population Proportion
The binomial distribution (the one that really applies) was approximated using the normal as long as np and nq were at least five. That is another way of saying the expected frequency of each category (success and failure) is at least five.
Population Variance
It was required that the population be normally distributed.
Correlation and Regression
The pairs of data had to have a bi-variate normal distribution.
Multinomial Experiment
The expected frequency of each category had to be at least five. This is analogous to approximating the binomial using the normal.
Independence
The expected frequency of each cell had to be at least five. This is analogous to approximating the binomial using the normal.

### Distributions

The distributions have normality in them somewhere, too.

Normal Distribution
Well, obviously this one requires normality.
Student's T Distribution
Had to be approximately normal. As the sample size increases, the student's t approaches the normal distribution.
Chi-squared Distribution
Required a normal population. There is another interesting relationship between the normal and chi-square distributions. If you take a critical value from normal distribution and square it, you will get the corresponding chi-square value with one degree of freedom, but twice the area in the tails.
Example: z(0.05)2 = 1.6452 = 2.706 = chi-square(1,0.10)
F Distribution
Since F is the ratio of two independent chi-squared variables divided by their respective degrees of freedom, and the chi-squares require a normal distribution, then the F distribution is also going to require a normal distribution.
Binomial Distribution
Obviously, the binomial doesn't require a normal population, but it can be approximated using a normal distribution if the expected frequency of each category is at least five.
Multinomial Distribution
Same as with the binomial, the multinomial can be approximated using the normal if the expected frequency of each category is at least five.

As stated in class and in the lecture notes ... your comprehension of the normal distribution is vital for success in the class.