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Chapter 3 The Normal Distributions BPS - 3rd Ed. Chapter 3 1 Density Curves Here is a histogram of vocabulary scores of 947 seventh graders The smooth curve drawn over the histogram is a mathematical model for the distribution. The mathematical model is called the density function. BPS - 3rd Ed. Chapter 3 2 Density Curves The areas of the shaded bars in this histogram represent the proportion of scores in the observed data that are less than or equal to 6.0. This proportion is equal to 0.303. BPS - 3rd Ed. Chapter 3 3 Areas under density curves The scale of the Y-axis of the density curve is adjusted so the total area under the curve is 1 The area under the curve to the left of 6.0 is shaded, and is equal to 0.293 This similar to the areas of the shaded bars in the prior slide! BPS - 3rd Ed. Chapter 3 4 Density Curves BPS - 3rd Ed. Chapter 3 5 Density Curves There are many types of density curves We are going to focus on a family of curves called Normal curves Normal curves are – bell-shaped – not too steep, not too fat – defined means & standard deviations BPS - 3rd Ed. Chapter 3 6 Normal Density Curves The mean and standard deviation computed from actual observations (data) are denoted by x and s, respectively. The mean and standard deviation of the distribution represented by the density curve are denoted by µ (“mu”) and (“sigma”), respectively. BPS - 3rd Ed. Chapter 3 7 Bell-Shaped Curve: The Normal Distribution standard deviation mean BPS - 3rd Ed. Chapter 3 8 The Normal Distribution Mean µ defines the center of the curve Standard deviation defines the spread Notation is N(µ,). BPS - 3rd Ed. Chapter 3 9 Practice Drawing Curves! Symmetrical around μ Infections points (change in slope, blue arrows) at ± σ BPS - 3rd Ed. Chapter 3 10 68-95-99.7 Rule for Any 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 BPS - 3rd Ed. Chapter 3 11 68-95-99.7 Rule for Any Normal Curve 68% - 95% µ + -2 µ +2 99.7% -3 BPS - 3rd Ed. µ Chapter 3 +3 12 68-95-99.7 Rule for Any Normal Curve BPS - 3rd Ed. Chapter 3 13 Men’s Height Example (NHANES, 1980) Suppose heights of men follow a Normal distribution with mean = 70.0 inches and standard deviation = 2.8 inches Shorthand: X ~ N(70, 2.8) X the variable ~ “distributed as” N(μ, σ) Normal with mean μ and standard deviation σ BPS - 3rd Ed. Chapter 3 14 Men’s Height Example 68-95-99.7 rule If X~N(70, 2.8) 68% between µ = 70.0 2.8 = 67.2 to 72.8 95% between µ 2 = 70.0 2(2.8) = 70.0 5.6 = 64.4 to 75.6 inches 99.7% between µ 3 = 70.0 3(2.8) = 70.0 8.4 = 61.6 and 78.4 inches BPS - 3rd Ed. Chapter 3 15 NHANES (1980) Height Example What proportion of men are less than 72.8 inches tall? (Note: 72.8 is one σ above μ on this distribution) 68% 16% ? -1 +1 84% BPS - 3rd Ed. (by 68-95-99.7 Rule) 70 Chapter 3 72.8 (height) 16 NHANES Height Example What proportion of men are less than 68 inches tall? ? 68 70 (height values) How many standard deviations is 68 from 70? BPS - 3rd Ed. Chapter 3 17 Standard Normal (Z) Distribution The Standard Normal distribution has mean 0 and standard deviation 1 We call this a Z distribution: Z~N(0,1) Any Normal variable x can be turned into a Z variable (standardized) by subtracting μ and dividing by σ: z BPS - 3rd Ed. x Chapter 3 18 Standardized Scores How many standard deviations is 68 from μ on X~N(70,2.8)? z = (x – μ) / σ = (68 70) / 2.8 = 0.71 The value 68 is 0.71 standard deviations below the mean 70 BPS - 3rd Ed. Chapter 3 19 Men’s Height Example (NHANES, 1980) What proportion of men are less than 68 inches tall? ? 68 70 -0.71 BPS - 3rd Ed. 0 (height values) (standardized values) Chapter 3 20 Table A in text: Standard Normal Table BPS - 3rd Ed. Chapter 3 21 Table A: Standard Normal Probabilities z .00 .01 .02 0.8 .2119 .2090 .2061 0.7 .2420 .2389 .2358 0.6 .2743 .2709 .2676 BPS - 3rd Ed. Chapter 3 22 Men’s Height Example (NHANES, 1980) What proportion of men are less than 68 inches tall? .2389 68 70 -0.71 BPS - 3rd Ed. 0 (height values) (standardized values) Chapter 3 23 Men’s Height Example (NHANES, 1980) What proportion of men are greater than 68 inches tall? Area under curve sums to 1, so Pr(X > x) = 1 – Pr(X < x), as shown below: 1.2389 = .2389 .7611 68 70 -0.71 BPS - 3rd Ed. 0 (height values) (standardized values) Chapter 3 24 Men’s Height Example (NHANES, 1980) How tall must a man be to place in the lower 10% for men aged 18 to 24? .10 ? 70 BPS - 3rd Ed. (height values) Chapter 3 25 Table A: Standard Normal Table Use Table A Look up the closest proportion in the table Find corresponding standardized score Solve for X (“un-standardize score”) BPS - 3rd Ed. Chapter 3 26 Table A: Standard Normal Proportion z .07 1.3 .0853 1.2 .1020 1.1 .1210 .08 .0838 .1003 .1190 .09 .0823 .0985 .1170 Pr(Z < -1.28) = .1003 BPS - 3rd Ed. Chapter 3 27 Men’s Height Example (NHANES, 1980) How tall must a man be to place in the lower 10% for men aged 18 to 24? .10 ? 70 -1.28 BPS - 3rd Ed. 0 (height values) (standardized values) Chapter 3 28 Observed Value for a Standardized Score “Unstandardize” z-score to find associated x : z x BPS - 3rd Ed. x z Chapter 3 29 Observed Value for a Standardized Score x = μ + zσ = 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 the population BPS - 3rd Ed. Chapter 3 30