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Chapter 3: Normal Distribution Density curves Normal distributions The rule Finding the normal proportions

3 Distribution of a Continuous Var. It is described by a density curve The density curve for a continuous var. X is a curve such that Proportion of X in between a and b is the area under it over the interval [a, b] The properties of a density curve: – It is always on or above the horizontal axis – The total area underneath it is one

4 Normal Distribution The “model” distribution of a continuous var. The normal density curve looks like: The standard normal density curve centers at 0 with standard deviation 1

5 Finding Normal Proportions X: a normal var. with mean  and standard deviation  P(X < a) = P() = see Table A (p ) z score

Finding Normal Proportions 1. State the problem and draw a picture 2. Calculate z scores and mark them on the picture 3. Use Table A

7 Example Suppose that the final scores of STAT1000 students follow a normal distribution with  = 70 and  = 10. What is the probability that a ST1000 student has final score 85 or above (grade A)? Between 75 and 85 (grade B)? Below 50 (F)?

Finding a Value given a Proportion 3 Steps: 1. State the problem and draw a Z curve picture 2. Use the table to find the z score 3. Unstandardize: find the corresponding x score Example: What is the first quartile of STAT 100 final score?

9 (for Bell-shaped distributions only)

10 Empirical Rule ( rule) If a variable X follows normal distribution, that is, all X values (the whole population) show bell-shaped, then: Mean(X) + 1*SD(X) covers 68% of possible X values Mean(X) + 2*SD(X) covers 95% of possible X values Mean(X) + 3*SD(X) covers 99.7% of possible X values

11 Empirical Rule ( rule) If the data (from a sample) of a variable X show bell-shaped, then: X + 1*S covers about 68% of possible X values X + 2*S covers about 95% of possible X values X + 3*S covers about 99.7% of possible X values

How to use Empirical Rule Find the range covering 68%, 95% or 99.7% of X values Check if X follows a normal distribution. 12