1 / 25

Chapter 3

Chapter 3. The Normal Distributions . Density Curves. Here is a histogram of vocabulary scores of n = 947 seventh graders The smooth curve drawn over the histogram is a mathematical model which represents t he density function of the distribution. Density Curves.

madeline
Télécharger la présentation

Chapter 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 3 The Normal Distributions Chapter 3

  2. Density Curves • Here is a histogram of vocabulary scores of n = 947 seventh graders • The smooth curve drawn over the histogram is a mathematical model which represents the density function of the distribution Chapter 3

  3. Density Curves • The shaded bars on this histogram corresponds to the scores that are less than 6.0 • This area represents is 30.3% of the total area of the histogram and is equal to the percentage in that range Chapter 3

  4. Area Under the Curve (AUC) • This figure shades area under the curve (AUC) corresponding to scores less than 6 • This also corresponds to the proportion in that range: AUC = proportion in that range Chapter 3

  5. Density Curves Chapter 3

  6. Mean and Median of Density Curve Chapter 3

  7. Normal Density Curves • Normal density curves are a family of bell-shaped curves • The mean of the density is denoted μ (“mu”) • The standard deviation is denoted σ (“sigma”) Chapter 3

  8. The Normal Distribution • Mean μ defines the center of the curve • Standard deviation σ defines the spread • Notation is N(µ,). Chapter 3

  9. Practice Drawing Curves! • The Normal curve is symmetrical around μ • It has infections (blue arrows) at μ ± σ Chapter 3

  10. The 68-95-99.7 Rule • 68% of AUC within μ± 1σ • 95% fall within μ± 2σ • 99.7% within μ± 3σ • Memorize! This rule applies only to Normal curves Chapter 3

  11. Application of 68-95-99.7 rule • Male height has a Normal distribution with μ= 70.0 inches and σ= 2.8 inches • Notation: Let X ≡ male height; X~ N(μ= 70, σ = 2.8) 68-95-99.7 rule • 68% in µ = 70.0  2.8 = 67.2 to 72.8 • 95% in µ 2 = 70.0  2(2.8) = 64.4 to 75.6 • 99.7% in µ 3 = 70.0  3(2.8) = 61.6 to 78.4 Chapter 3

  12. 68% 68% (by 68-95-99.7 Rule) (total AUC = 100%) ? 16% 16% -1 +1 70 72.8 (height) 84% Application: 68-95-99.7 Rule What proportion of men are less than 72.8 inches tall? μ + σ = 70 + 2.8 = 72.8 (i.e., 72.8 is one σ above μ) Therefore, 84% of men are less than 72.8” tall. Chapter 3

  13. ? 68 70 (height values) Finding Normal proportions What proportion of men are less than 68” tall? This is equal to the AUC to the left of 68 on X~N(70,2.8) To answer this question, first determine the z-score for a value of 68 from X~N(70,2.8) Chapter 3

  14. Z score • The z-score tells you how many standard deviation the value falls below (negative z score) or above (positive z score) mean μ • The z-score of 68 when X~N(70,2.8) is: Thus, 68 is 0.71 standard deviations below μ. Chapter 3

  15. ? 68 70 (height values) Example: z score and associate value -0.71 0 (z values) Chapter 3

  16. Standard Normal Table Use Table A to determine the cumulative proportion associated with the z score See pp. 79 – 83 in your text! Chapter 3

  17. Normal Cumulative Proportions (Table A) .01 0.7 .2389 Thus, a z score of −0.71 has a cumulative proportion of .2389 Chapter 3

  18. 68 70 (height values) -0.71 0 (z scores) Normal proportions The proportion of mean less than 68” tall (z-score = −0.71 is .2389: .2389 Chapter 3

  19. 68 70 (height values) -0.71 0 (z values) Area to the right (“greater than”) Since the total AUC = 1: AUC to the right = 1 – AUC to left Example: What % of men are greater than 68” tall? 1.2389 = .7611 .2389 Chapter 3

  20. Normal proportions “The key to calculating Normal proportions is to match the area you want with the areas that represent cumulative proportions. If you make a sketch of the area you want, you will almost never go wrong. Find areas for cumulative proportions … from [Table A] (p. 79)” Follow the “method in the picture” (see pp. 79 – 80) to determine areas in right tails and between two points Chapter 3

  21. Finding Normal values We just covered finding proportions for Normal variables. At other times, we may know the proportion and need to find the Normal value. Method for finding a Normal value: 1. State the problem 2. Sketch the curve 3. Use Table A to look up the proportion & z-score 4. Unstandardize the z-score with this formula Chapter 3

  22. .10 ? 70 (height) State the Problem & Sketch Curve Problem: How tall must a man be to be taller than 10% of men in the population? (This is the same as asking how tall he has to be to be shorter than 90% of men.) Recall X~N(70, 2.8) Chapter 3

  23. Table AFind z score for cumulative proportion ≈.10 .08 1.2 .1003 zcum_proportion = z.1003 = −1.28 Chapter 3

  24. .10 ? 70 (height values) Visual Relationship Between Cumulative proportion and z-score -1.28 0 (Z value) Chapter 3

  25. Unstandardize • x = μ + z∙σ = 70 + (-1.28 )(2.8) = 70 + (3.58) = 66.42 • Conclude: A man would have to be less than 66.42 inches tall to place him in the lowest 10% of heights Chapter 3

More Related