1. The future is not come yet and the past is gone. Uphold our loving kindness at this instant, and be committed to our duties and responsibilities right now.

2. Chapter 3: Normal Distribution Density curves Normal distributions The 68-95-99.7 rule Finding the normal proportions

3. 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. 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. 5 Finding Normal Proportions X: a normal var. with mean  and standard deviation  P(X < a) = P() = see Table A (p. 690-691) z score

6. 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. 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)?

8. 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. 9 (for Bell-shaped distributions only)

10. 10 Empirical Rule (68-95-99.7 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. 11 Empirical Rule (68-95-99.7 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

12. 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