Chapter 3

1. Chapter 3 The Normal Distributions Chapter 3

2. Z-Score Explained • http://www.youtube.com/watch?v=AT-HH0W_swA&feature=related Basics of Using the Std Normal Table • http://www.youtube.com/watch?v=y6sbghmHwQA&feature=related Normal Distribution & Z-score • http://www.youtube.com/watch?v=mai23vW8uFM&feature=related Chapter 3

3. We’ll Learn The Topics • Review Histogram • Density Curve • Normal Distribution • 68 – 95 – 99.7 Rule • Z-score • Standard Normal Distribution Chapter 3

4. Density Curves Example: here is a histogram of vocabulary scores of 947 seventh graders. • We can describe the histogram with a smooth curve, a bell- shaped curve. • It corresponding to a normal distribution Model. Chapter 3

5. Density Curves Example: the areas of the shaded bars in this histogram represent the proportion of scores that are less than or equal to 6.0. This proportion in the observed data is equal to 0.303. Chapter 3

6. Density Curves ■ now the area under the smooth curve to the left of 6.0 is shaded. ■ The scale is adjusted, the total area under the curve is exactly 1, this curve is called a density curve. ■ The proportion of the area to the left of 6.0 is now equal to 0.293. Chapter 3

7. Density Curves • Always on or above the horizontal axis • Have area exactly 1 underneath curve • Display the bell-shaped pattern of a distribution • A histogram becomes a density curve if the scale is adjusted so that the total area of the bars is 1. Chapter 3

8. Mean & Standard Deviation • The mean and standard deviation computed from actual observations (data) are denoted by and s,respectively • The mean and standard deviation of the distribution represented by the density curve are denoted by µ (“mu”) and  (“sigma”), respectively. • The mean of a density curve is the "balance point" of the curve. Chapter 3

9. standard deviation mean Bell-Shaped Curve:The Normal Distribution Chapter 3

10. The Normal Distribution ■Knowing the mean (µ)and standard deviation () allows us to make various conclusions about Normal distributions. ■ Notation: N(µ,). Chapter 3

11. 68-95-99.7 Rule forAny Normal Curve • 68% of the observations fall within one standard deviation of the mean • 95% of the observations fall within two standard deviations of the mean • 99.7% of the observations fall within three standard deviations of the mean Chapter 3

12. 68% 95% - µ + -2 µ +2 99.7% -3 +3 µ 68-95-99.7 Rule forAny Normal Curve Chapter 3

13. 68-95-99.7 Rule forAny Normal Curve Chapter 3

14. Health and Nutrition Examination Study of 1976-1980 • Heights of adult men, aged 18-24 • mean: 70.0 inches • standard deviation: 2.8 inches • heights follow a normal distribution, so we have that heights of men are N(70, 2.8). Chapter 3

15. Health and Nutrition Examination Study of 1976-1980 • 68-95-99.7 Rule for men’s heights • 68% are between 67.2 and 72.8 inches [ µ = 70.0  2.8 ] • 95% are between 64.4 and 75.6 inches [ µ 2 = 70.0  2(2.8) = 70.0  5.6 ] • 99.7% are between 61.6 and 78.4 inches [ µ 3 = 70.0  3(2.8) = 70.0  8.4 ] Chapter 3

16. 68% (by 68-95-99.7 Rule) ? 16% -1 +1 70 72.8 (height values) ? = 84% Health and Nutrition Examination Study of 1976-1980 • What proportion of men are less than 72.8 inches tall? Chapter 3

17. Standard Normal Distribution • Z – Score • The standard Normal distributionN(0,1) is the Normal distributionhas a mean of zero and a standard deviation of one • Normal distributions can be transformed to standard normal distributions by Z-score Chapter 3

18. ? 68 70 (height values) Health and Nutrition Examination Study of 1976-1980 • What proportion of men are less than 68 inches tall? How many standard deviations is 68 from 70? Chapter 3

19. Standardized Scores • standardized score (Z-score) = (observed value minus mean) / (std dev) [ = (68 - 70) / 2.8 = -0.71 ] • The value 68 is 0.71 standard deviations below the mean 70. Chapter 3

20. ? 68 70 (height values) Health and Nutrition Examination Study of 1976-1980 • What proportion of men are less than 68 inches tall? -0.71 0 (standardized values) Chapter 3

21. Table A:Standard Normal Probabilities • See pages 464-465 in text for Table A.(the “Standard Normal Table”) • Look up the closest standardized score (z) in the table. • Find the probability (area) to the left of the standardized score. Chapter 3

22. Table A:Standard Normal Probabilities Chapter 3

23. Table A:Standard Normal Probabilities .01 0.7 .2389 Chapter 3

24. 68 70 (height values) -0.71 0 (standardized values) Health and Nutrition Examination Study of 1976-1980 • What proportion of men are less than 68 inches tall? .2389 Chapter 3

25. 68 70 (height values) -0.71 0 (standardized values) Health and Nutrition Examination Study of 1976-1980 • What proportion of men are greater than 68 inches tall? 1.2389 = .7611 .2389 Chapter 3

26. .10 ? 70 (height values) Health and Nutrition Examination Study of 1976-1980 • How tall must a man be to place in the lower 10% for men aged 18 to 24? Chapter 3

27. Table A:Standard Normal Probabilities • See pages 464-465 in text for Table A. • Look up the closest probability (to .10 here) in the table. • Find the corresponding standardized score. • The value you seek is that many standard deviations from the mean. Chapter 3

28. Table A:Standard Normal Probabilities .08 1.2 .1003 Chapter 3

29. .10 ? 70 (height values) Health and Nutrition Examination Study of 1976-1980 • How tall must a man be to place in the lower 10% for men aged 18 to 24? -1.28 0 (standardized values) Chapter 3

30. Observed Value for a Standardized Score • Need to “reverse” the z-score to find the observed value (x) : • observed value = mean plus [(standardized score)  (std dev)] Chapter 3

31. Observed Value for a Standardized Score • observed value = mean plus [(standardized score)  (std dev)] = 70 + [(-1.28 ) (2.8)] = 70 + (3.58) = 66.42 • A man would have to be approximately 66.42 inches tall or less to place in the lower 10% of all men in the population. Chapter 3

32. Chapter 3

33. Chapter 3

34. The Entry in Table A • Using random variable z to get the entrance in Table A. • Variable z is z-score which follows the standard normal distribution N(0, 1) • Z-score: • When search entry for a z value ♫look up the most left column first, locate the most close value to z value ♫ look up the top row to locate the 2th decimal place for a z value Chapter 3

35. The Entry in Table A • Table A’s entry is an area underneath the curve, to the left of z • Table A’s entry is a percent of the whole area, to the left of z-score • Table A’s entry is a probability, corresponding to the z-score value. • Math formula: P (z ≤ z0) = 0.xxxx P (z ≤ -0.71) = 0.2389 Chapter 3

36. Problem type I • If z ≈ N(0, 1), P (z ≤ z0) = ? • By checking the Table A, find out the answer. • For type I problem, check the table and get the answer directly. For example, P (z ≤ -0.71) = 0. 2389 Chapter 3

37. Problem type II • If z ≈ N(0, 1), P (z ≥ z0) = ? • This type’s problem, cannot check the table directly. Using the following operation. • P (z ≥ z0) = 1 - (z ≤ z0) • For example, p (z ≥ -0.71) = ? ◙ P (z ≥ -0.71) = 1 - (z ≤ - 0.71) = 1 – 0.2389 = 0.7611 Chapter 3

38. 68 70 (height values) -0.71 0 (standardized values) Problem Type II 0.7611 0.2389 Chapter 3

39. Problem Type III • If z ≈ N(0, 1), P ( z2 ≤ z ≤ z1) = ? random variable z is between two numbers • Look up z1 →P1 • Look up z2 →P2 • The result is: P ( z2 ≤ z ≤ z1) = P1 -P2 • For example, P ( -1.4 ≤ z ≤ 1.3) = ? look up 1.3 P1 =0.9032 look up -1.4 P2 = 0.0808 P ( -1.4 ≤ z ≤ 1.3) = 0.9032 – 0.0808 = 0.8224 Chapter 3

40. Steps Summary • Write down the normal distribution N(µ,) for observation data set • Locate the specific observation value X0 • Transform X0 to be Z0 by z-score formula • Check table A using random variable Z0 tofind out table entry P(z ≤ z0) • If is problem type I, the result is P(z ≤ z0) • If is problem type II, the result is: P (z ≥ z0) =1- P(z ≤ z0) • If is problem type III, the result is: P ( z2 ≤ z ≤ z1) = P1 -P2 P1 =P(z ≤ z1), P2 =P(z ≤ z2) Chapter 3