Normal Distribution Section 2.2

Objectives  Introduce the Normal Distribution  Properties of the Standard Normal Distribution  Use Normal Distribution in an inferential fashion

Theoretical Distribution  Empirical distributions based on data  Theoretical distribution based on mathematics  derived from model or estimated from data

Normal Distribution Why are normal distributions so important?  Many variables are commonly assumed to be normally distributed in the population Height, weight, IQ scores, ACT scores, etc.  If a variable is approximately normally distributed we can make inferences about values of that variable

Normal Distribution  Since we know the shape of the curve, we can calculate the area under the curve  The percentage of that area can be used to determine the probability that a given value could be pulled from a given distribution The area under the curve tells us about the probability- in other words we can obtain a p-value for our result (data) by treating it as a normally distributed data set.

Key Areas under the Curve  For normal distributions + 1 SD ~ 68% + 2 SD ~ 95% + 3 SD ~ 99.7%

Standard Normal Distributions Standard Normal Distribution – N( ,  )  We agree to use the standard normal distribution  Roughly symmetric   =0   =1

Recall: Z-score If we know the population mean and population standard deviation, for any value of X we can compute a z-score by subtracting the population mean and dividing the result by the population standard deviation

Proportions  Total area under the curve is 1  The area in red is equal to p(z > 1)  The area in blue is equal to p(-1< z <0)  Since the properties of the normal distribution are known, areas can be looked up on tables or found with a calculator.

Suppose Z has standard normal distribution Find 0 < Z < 1.23

Find < Z < 0

Find Z > 0.78

Z is standard normal Calculate -1.2 < Z < 0.78

Work time...  What is the area for scores less than z = -1.5?  What is the area between z =1 and 1.5?  What z-score cuts off the highest 30% of the distribution?  What two z-scores enclose the middle 50% of the distribution?