1. Statistics 270 - Lecture 14

2. Example Consider a rv, X, with pdf Sketch pdf

3. Example cdf of X: Find E(X)

4. Normal Distributions Common continuous density is the normal distribution It is symmetric, bell-shaped and uni-modal Denoted

5. Normal Distributions Density: cdf:

7. What happens if mean is changed? What happens if standard deviation is changed?

10. Standard Normal The standard normal distribution is a particular normal distributrion X~N(0,1) pdf: Have table of cumulative probabilities for standard normal (Table A-3)

11. Example Suppose Z has a standard nomral distribution. Find: P(Z<1.96)= P(Z<3.02)= P(Z>3.03)=

12. Example Suppose Z has a standard nomral distribution. Find: P(Z<3.025)=

13. Standardizing If X is any R.V., the standardized variable, Z, has mean 0 and standard deviation 1

14. Distribution of scores on a standardized test can be approximated by a normal distribution with mean of 500 and standard deviation of 100. Find probability that a randomly selected student scores: Over 650 Between 325 and 675 What proportion of students score better than 680?

15. 68-95-99.7 Rule For a random variable, X, that is normally distributed with mean,  and standard deviation,  : 68% of the observations will fall within 1 standard deviation of the mean 95% of the observations will fall within 2 standard deviation of the mean 99.7% of the observations will fall within 3 standard deviation of the mean

16. Example (page 65) The distribution of heights of young women aged 20-29 is approximately Normally distributed with mean 64 inches and standard deviation 2.7 inches Between what heights do 95% of the heights of young women fall? What percent of young women are taller than 61.3 inches?