2.2: Normal Distributions

1. 2.2: Normal Distributions Note on Uniform Distributions

2. 3 Reasons why we like Normal Distributions • Good descriptions of real data (ex: SATs, psychological tests, characteristics of populations…) • Good approximations to results of many kinds of chance outcomes. • Many inference procedures work well for “roughly symmetrical” distributions. • Many data sets tend to be mound-shaped (characteristics of biological populations) • TI83: student heights, L1, graph

3. Normal Distributions • Described by giving its mean and std. deviation • controls the spread of a normal curve. Figure shows curve w/different values of . • Changing w/o changing moves the curve along the horizontal axis w/o changing spread.

4. Locating the standard deviation by eyeballing the curve: “Inflection Points”

5. Common Properties of Normal Curves • They all have inflection points (where change of curvature takes place). • E. rule only provides an approximate value for the proportion of observations that fall within 1, 2, or 3 std. devs of the mean.

6. Example #1 • Suppose that taxicabs in NYC are driven an average of 75,000 miles per year with a standard deviation of 12,000 miles. What information does the empirical rule tell us?

7. 2 Normal curves What do you notice about their means? What do you notice about their standard deviations?

11. Find the proportion of observations from the standard normal distribution that are less than 2.22. That is: Find P (z < 2.22) Finding Areas to the Left

12. Find the proportion of observations from the standard normal distribution that are greater than -2.15. That is: find P (z > -2.15) Finding Areas to the Right

13. Table A Practice Use Table A to find the proportion of observations from a standard Normal distribution that falls in each of the following regions. In each case, sketch a standard Normal curve and shade the area representing the region. • Z is less than or equal to -2.25 • Z is greater than or equal to -2.25 • Z > 1.77 • -2.25 < z < 1.77

14. Example • The mean of women is 64.5 inches, and the standard deviation is 2.5 inches. What proportion of all young women are less than 68 inches tall?

15. Example • The level of cholesterol in the blood is important because high cholesterol levels may increase the risk of heart disease. The distribution of blood cholesterol levels in a large population of people of the same age and sex is roughly normal. For 14 year old boys, the mean is 170 mg/dl and the standard deviation is 30 mg/ . Levels above 240 mg/dl may require medical attention. What percent of 14-year-old boys have more than 240 mg/dl of cholesterol?

17. What percent of 14 year old boys have between 170 and 240 mg/dl?

18. Finding a value given a proportion • Use Table A backwards! 1) Find the given proportion in the body of the table 2) Read the corresponding z-score 3) Unstandardize to get the observed (x) value. Voila!

19. Example • Scores on the SAT verbal test in recent years follow approximately the N(505,110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT?

21. Special Note…. • X is greater than is the same as x is greater than or equal to. • That is, there is no area under the curve where x = 240. There may be a boy with an exact cholesterol level of 240, but there is no area under the curve at an exact point. • The normal distribution is therefore an approximation – not a description of every detail in the exact data.

22. Normal Probability Plot • TI83 • Don’t overreact to minor wiggles in the plot • Normality cannot be assumed if there is skewness or outliers (don’t use Normal distribution if these things occur)!