The Standard Normal Distribution

1. The Standard Normal Distribution Section 2.2.1

2. Starter • Weights of adult male Norwegian Elkhounds are N(42, 2) pounds. • What weight would represent the 16th percentile?

3. Today’s Objectives • State mean and standard deviation for The Standard Normal Distribution. • Given a raw score from a normal distribution, find the standardized “z-score”. • Use the Table of Standard Normal Probabilities to find: • the area between given borders of the Standard Normal curve. • The raw score (x value) that leads to a given area.

4. Standardizing Observations • All normal distributions have fundamentally the same shape. • If we measure the x axis in units of size σ about a center of 0, then they are all exactly the same curve. • This is called the Standard Normal Curve • To standardize observations, we change from x values (the raw observations) to z values (the standardized observations) by the formula:

5. The Standard Normal Distribution • Notice that the z-score formula always subtracts μ from each observation. • So the mean is always shifted to zero • Also notice that the shifted values are divided by σ, the standard deviation. • So the units along the z-axis represent numbers of standard deviations • Thus the Standard Normal Distribution is always N(0,1).

6. Example 2.3 (p 84) • The heights of young women are: N(64.5, 2.5) • Use the formula to find the z-score of a woman 68 inches tall. • A woman’s standardized height is the number of standard deviations by which her height differs from the mean height of all young women.

7. Normal Distribution Calculations • What proportion of all young women are less than 68 inches tall? (Examples 2.4 & 2.5, p 86) • Notice that this does not fall conveniently on one of the σ borders, so the empirical rule does not help! • We already found that 68 inches corresponds to a z-score of 1.4 • So what proportion of all standardized observations fall to the left of z = 1.4? • Since the area under the Standard Normal Curve is always 1, we can ask instead what is the area under the curve and to the left of z=1.4 • For that, we need a table.

8. The Standard Normal Table • Find Table A on pages 13 & 14 of the handout • It is also in your textbook inside the front cover • Z-scores (to the nearest tenth) are in the left column • The other 10 columns round z to the nearest hundredth • Find z = 1.4 in the table and read the area • You should find area to the left = .9192 • So the proportion of observations less than z = 1.4 is about 92% • Now put the answer in context: “About 92% of all young women are 68 inches tall or less.”

9. What about area above a value? • Still using the N(64.5, 2.5) distribution, what proportion of young women have a height of 61.5 inches or taller? • Z = (61.5 – 64.5)/2.5 = -1.2 • From Table A, area to the left of -1.2 =.1151 • So area to the right = 1 - .1151 = .8849 • So about 88% of young women are 61.5” tall or taller.

10. What about area between two values? • What proportion of young women are between 61.5” and 68” tall? • We already know 68” gives z = 1.4 and area to the left of .9192 • We also know 61.5” gives z = -1.2 and area to the left of .1151 • So just subtract: .9192 - .1151 = .8041 • So about 80% of young women are between 61.5” and 68” tall • Remember to write your answer IN CONTEXT!!!

11. Given a proportion, find the observation x (Ex. 2.8, p 90) • SAT Verbal scores are N(505, 110). How high must you score to be in the top 10%? • If you are in the top 10%, there must be 90% below you (to the left). • Find .90 (or close to it) in the body of Table A. What is the z-score? • You should have found z = 1.28 • Now solve the z definition equation for x • So you need a score of at least 646 to be in the top 10%.

12. Today’s Objectives • State mean and standard deviation for The Standard Normal Distribution. • Given a raw score from a normal distribution, find the standardized “z-score”. • Use the Table of Standard Normal Probabilities to find: • the area between given borders of the Standard Normal curve. • The raw score (x value) that leads to a given area.

13. Homework • Read pages 83 – 91 • Do problems 22 – 25