HS 67 - Intro Health Stat The Normal Distributions Thursday, May 16, 2019 Thursday, May 16, 2019 Chapter 11 The Normal Distributions Note: Normal distributions are the most popular probability models in statistics. Other popular models include binomial distributions, t distributions, etc. 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 1 1

Recall: Rules About Density Curves HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Recall: Rules About Density Curves 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 2 2

Recall the relationship between AUCs and probabilities HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Recall the relationship between AUCs and probabilities Area Under Curve = probability for that range 30% of students had scores ≤ 6 30% of area under the curve (AUC) is shaded 30% 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 3 3

The Normal pdf is: A family of related bell-shaped density HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 The Normal pdf is: A family of related bell-shaped density Each family member has a different μ (mean) and σ (standard deviation) μ  location σ  spread There are an infinite number of Normal density curves (curve changes every time µ and/or σchanges) Notation: X~N(µ,σ) is read “X is distributed as …” 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 5 5

Draw Normal curves accurately! Symmetrical around μ Infection points at μ ± σ Horizontal asymptotes Drawing a half circle is not acceptable 5/16/2019

The 68-95-99.7 rule applies only to Normal densities HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 68-95-99.7 Rule μ ± 1σ contains 68% of AUC μ ± 2σ contains 95% of AUC μ ± 3σ contains 99.7% of AUC The 68-95-99.7 rule applies only to Normal densities 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 7 7

HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 68-95-99.7 Rule - Example Assume male height (X) has Normal distribution with μ = 70.0 inches and σ = 2.8 inches Notation: X~ N(70, 2.8) µ   = 70.0  2.8 = 67.2 to 72.8  68% of men between 67.2 and 72.8 inches tall µ  2 = 70.0  2(2.8) = 64.4 to 75.6 (inches)  95% in this range µ  3 = 70.0  3(2.8) = 61.6 to 78.4 (inches)  99.7% in this range BPS Chapter 3 Normal DistributionsBPS Chapter 3 8 8

68-95-99.7 Rule example (cont.) Height in inches 70 72.8 75.6 78.4 HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 68-95-99.7 Rule example (cont.) 70 Height in inches 72.8 75.6 78.4 67.2 64.4 61.6 5/16/2019 9 BPS Chapter 3 Normal DistributionsBPS Chapter 3 9 9

What proportion of men are less than 72.8" tall? HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Example 2 (Male Height) What proportion of men are less than 72.8" tall? X~ N(70, 2.8) By 68-95-99.7 rule 68% Total AUC =100%  32% split evenly in tails 16% 16% 16% 16% 16% + 68% = 84% 84% of AUC to the left of 72.8 5/16/2019 10 BPS Chapter 3 Normal DistributionsBPS Chapter 3 10 10

Finding Normal Probabilities: Method Draw an reasonably accurate Normal curve with landmarks at µ and µ ±σ Place values on the curve & shade relevant area Standardize values (i.e., z scores) Use Table A to determine the AUC 5/16/2019

Finding Normal Probabilities: Draw & Shade HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Finding Normal Probabilities: Draw & Shade What proportion of men are less than 68” tall? ? 68 70 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 12 12

Finding Normal Probabilities: Standardize HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Finding Normal Probabilities: Standardize In X~N(70, 2.8), the value 68 has: This indicates that 68 is 0.71 standard deviations below the mean 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 13 13

Finding Normal Probabilities: Standardize HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Finding Normal Probabilities: Standardize Proportion of men less than 68” tall = Pr(Z < -0.71): 68 70 (height) -0.71 0 (z score) 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 14 14

Standard Normal “Z” Table HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Standard Normal “Z” Table Notation: zp where p ≡ cumulative probability = AUC to left p Z is the Normal random variable with µ = 0 and σ= 1 Z~N(0,1) There is only one Standard Normal Z variable Z table in text (pp. 677-8) and in “Formulas” handout 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 15 15

Finding Normal Probabilities: Table A HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Finding Normal Probabilities: Table A z .00 .02 0.8 .2119 .2090 .2061 .2420 .2358 0.6 .2743 .2709 .2676 .01 0.7 .2389 z score of −0.71 → AUC to left is .2389 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 16 16

Conclude: .2389 (23.89%) of men are less than 68 inches in height HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Pr(Z < −0.7) as an Image Conclude: .2389 (23.89%) of men are less than 68 inches in height From Table A .2389 68 70 (height) -0.71 0 (z score) 5/16/2019 17 BPS Chapter 3 Normal DistributionsBPS Chapter 3 17 17

Pr(Z > z)  AUC to the right HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Pr(Z > z)  AUC to the right Ask: What proportion of men greater than 68” tall? “Greater than” = “area to right of”= 1 – (AUC to left) 1 – .2389 = .7611 .2389 68 70 (height) -0.71 0 (z score) 5/16/2019 18 BPS Chapter 3 Normal DistributionsBPS Chapter 3 18 18

AUC Between Two Points Total AUC = 1. Therefore: Pr(a < Z < b) = Pr(Z < b) – Pr(Z < a) .6915 .1003 -1.28 0.5 0.5 -1.28 Pr(−1.28 < Z < 0.5) = Pr(Z < 0.5) – Pr(Z < −1.28) = .6915 – .1003 = .5912

Finding z percentiles and associated values HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Finding z percentiles and associated values At times, we must find the z value or X value that is associated with a given probability. To do this: 1. State the problem 2. Sketch the curve 3. Look up the related z-score is Z table 4. If you need to know X, “unstandardize” z with this formula: 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 20 20

State & Sketch ? 0 (z equivalent) z.10 70 (height) 70 (height) HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 State & Sketch How tall must a man be to be taller than 10% of men (i.e., what is the 10th percentile of heights)? Given: X~N(70, 2.8) .10 .10 ? 70 (height) 70 (height) 0 (z equivalent) z.10 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 21 21

Find z score in Table A  z.10 = -1.28 z .07 .09 1.3 .0853 .0838 HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Find z score in Table A z .07 .09 1.3 .0853 .0838 .0823 .1020 .0985 1.1 .1210 .1190 .1170 .08 1.2 .1003  z.10 = -1.28 5/16/2019 BPS Chapter 3 Normal DistributionsBPS Chapter 3 22 22

Visual Relationship Notation: z.1003 = −1.28 X~N(70, 2.8) .10 HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Visual Relationship Notation: z.1003 = −1.28 AUC to left z-score X~N(70, 2.8) .10 70 (height) 70 Height (inches) ? Height (z score) −1.28 5/16/2019 Chapter 3 BPS Chapter 3 Normal DistributionsBPS Chapter 3 23 23

Unstandardize & Conclude HS 67 - Intro Health Stat HS 67 - Intro Health Stat Thursday, May 16, 2019 Thursday, May 16, 2019 Unstandardize & Conclude x = μ + z∙σ = 70 + (-1.28 )(2.8) = 70 + (3.58) = 66.42 Conclude: A man who is 66.42 inches in height is taller than 10% of men We can also conclude that he is shorter than (1 – 10%) or 90% of men 5/16/2019 Chapter 3 BPS Chapter 3 Normal DistributionsBPS Chapter 3 24 24