HS 67 - Intro Health Stat The Normal Distributions Saturday, April 15, 2017 Saturday, April 15, 2017 Chapter 3 The Normal Distributions 4/15/2017 Chapter 3 BPS Chapter 3 1

HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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 4/15/2017 Chapter 3 BPS Chapter 3 2

HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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 4/15/2017 Chapter 3 BPS Chapter 3 3

Area Under the Curve (AUC) HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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 4/15/2017 Chapter 3 BPS Chapter 3 4

Density Curves 4/15/2017 Chapter 3 HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 Density Curves 4/15/2017 Chapter 3 BPS Chapter 3 5

Mean and Median of Density Curve 4/15/2017 Chapter 3

HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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”) 4/15/2017 Chapter 3 BPS Chapter 3 7

The Normal Distribution HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 The Normal Distribution Mean μ defines the center of the curve Standard deviation σ defines the spread Notation is N(µ,). 4/15/2017 Chapter 3 BPS Chapter 3 8

Practice Drawing Curves! The Normal curve is symmetrical around μ It has infections (blue arrows) at μ ± σ 4/15/2017 Chapter 3

The 68-95-99.7 Rule 68% of AUC within μ ± 1σ 95% fall within μ ± 2σ HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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 4/15/2017 Chapter 3 BPS Chapter 3 10

HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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 4/15/2017 Chapter 3 BPS Chapter 3 11

? Application: 68-95-99.7 Rule 70 72.8 (height) 84% 68% 16% 16% 68% HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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) ? 70 72.8 (height) 68% (by 68-95-99.7 Rule) -1 +1 16% 16% 68% (total AUC = 100%) 84% Therefore, 84% of men are less than 72.8” tall. 4/15/2017 Chapter 3 BPS Chapter 3 12

Finding Normal proportions HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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) ? 68 70 (height values) To answer this question, first determine the z-score for a value of 68 from X~N(70,2.8) 4/15/2017 Chapter 3 BPS Chapter 3 13

Thus, 68 is 0.71 standard deviations below μ. HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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. 4/15/2017 Chapter 3 BPS Chapter 3 14

Example: z score and associate value HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 Example: z score and associate value 68 70 (height values) ? -0.71 0 (z values) 4/15/2017 Chapter 3 BPS Chapter 3 15

Standard Normal Table See pp. 79 – 83 in your text! HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 Standard Normal Table Use Table A to determine the cumulative proportion associated with the z score See pp. 79 – 83 in your text! 4/15/2017 Chapter 3 BPS Chapter 3 16

Normal Cumulative Proportions (Table A) HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 Normal Cumulative Proportions (Table A) z .00 .02 0.8 .2119 .2090 .2061 .2420 .2358 0.6 .2743 .2709 .2676 .01 0.7 .2389 Thus, a z score of −0.71 has a cumulative proportion of .2389 4/15/2017 Chapter 3 BPS Chapter 3 17

The proportion of mean less than 68” tall (z-score = −0.71 is .2389: HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 Normal proportions The proportion of mean less than 68” tall (z-score = −0.71 is .2389: -0.71 0 (z scores) 68 70 (height values) .2389 4/15/2017 Chapter 3 BPS Chapter 3 18

Area to the right (“greater than”) HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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? -0.71 0 (z values) 68 70 (height values) 1.2389 = .7611 .2389 4/15/2017 Chapter 3 BPS Chapter 3 19

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 4/15/2017 Chapter 3

HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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 4/15/2017 Chapter 3 BPS Chapter 3 21

State the Problem & Sketch Curve HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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) .10 ? 70 (height) 4/15/2017 Chapter 3 BPS Chapter 3 22

Table A Find z score for cumulative proportion ≈.10 HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 Table A Find z score for cumulative proportion ≈.10 z .07 .09 1.3 .0853 .0838 .0823 .1020 .0985 1.1 .1210 .1190 .1170 .08 1.2 .1003 zcum_proportion = z.1003 = −1.28 4/15/2017 Chapter 3 BPS Chapter 3 23

Visual Relationship Between Cumulative proportion and z-score HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 Visual Relationship Between Cumulative proportion and z-score .10 ? 70 (height values) -1.28 0 (Z value) 4/15/2017 Chapter 3 BPS Chapter 3 24

Unstandardize x = μ + z∙σ = 70 + (-1.28 )(2.8) = 70 + (3.58) = 66.42 HS 67 - Intro Health Stat Saturday, April 15, 2017 Saturday, April 15, 2017 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 4/15/2017 Chapter 3 BPS Chapter 3 25