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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 z X 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 -1.57 < 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?