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Chapter 13 Normal Distributions Chapter 13 1 Thought Question 1 Birth weights of babies born in the United States follow, at least approximately, a bell- shaped curve. What does that mean? Thought Question 2 What does it mean if a person’s SAT score falls at the 20th percentile for all people who took the test? Chapter 13 2 Thought Question 3 A study in 1976-80 found that males (ages 18-24) have a mean height of 70 inches and a standard deviation of 2.8 in., while females (ages 18-24) have a mean height of 65 in. and a standard deviation of 2.5 in. A “standardized score” is the number of standard deviations an individual falls above or below the mean for the whole group (is positive for values above the mean, and negative for those below the mean). Thus, a man who is 72.8 inches tall has a standardized score of 1. What is the standardized score for your height? Chapter 13 3 Thought Question 4 Many measurements in nature tend to follow a similar pattern. The pattern is that most of the individual measurements take on values that are near the average, with fewer and fewer measurements taking on values that are farther from the average in either direction. Describe what shape the distribution of such measurements would have. Chapter 13 4 Bell-Shaped Curve: The Normal Distribution of Population Values Chapter 13 5 Asymmetric Distributions of the Population Values Chapter 13 6 The Normal Distribution standard deviation mean Chapter 13 7 With the Mean and Standard Deviation of the Normal Distribution We Can Determine: What proportion of individuals fall into any range of values At what percentile a given individual falls, if you know their value What value corresponds to a given percentile Chapter 13 8 Empirical Rule for Any Normal Curve 68% of the values fall within one standard deviation of the mean 95% of the values fall within two standard deviations of the mean 99.7% of the values fall within three standard deviations of the mean “68-95-99.7 Rule” Chapter 13 9 Empirical Rule for Any Normal Curve 68% 95% -1sd +1sd -2 sd +2 sd 99.7% -3 sd +3 sd Chapter 13 10 Health and Nutrition Examination Study of 1976-1980 (HANES) Heights of adults, ages 18-24 – women mean: 65.0 inches standard deviation: 2.5 inches – men mean: 70.0 inches standard deviation: 2.8 inches Chapter 13 11 Health and Nutrition Examination Study of 1976-1980 (HANES) Empirical Rule – women 68% are between 62.5 and 67.5 inches [mean 1 std dev = 65.0 2.5] 95% are between 60.0 and 70.0 inches 99.7% are between 57.5 and 72.5 inches – men 68% are between 67.2 and 72.8 inches 95% are between 64.4 and 75.6 inches 99.7% are between 61.6 and 78.4 inches Chapter 13 12 Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of men are less than 72.8 inches tall? 68% (by Empirical Rule) 32% / 2 = 16% ? -1 +1 standard 70 72.8 (height values) deviations: -3 ? -2 = 84% -1 0 1 2 3 height values: 61.6 64.4 67.2 70 72.8 75.6 78.4 Chapter 13 13 Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of men are less than 68 inches tall? ? standard 68 70 (height values) deviations: -3 -2 -1 0 1 2 3 height values: 61.6 64.4 67.2 70 72.8 75.6 78.4 Chapter 13 14 Standardized Scores How many standard deviations is 68 from 70? standardized 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 13 15 Standardized Scores standardized score = (observed value minus mean) / (std dev) z is the standardized score x is the observed value m is the population mean s is the population standard deviation x-m z s Chapter 13 16 Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of men are less than 68 inches tall? ? 68 70 (height values) -0.71 0 (standardized values) Chapter 13 17 Table B: Percentiles of the Standardized Normal Distribution See Table B (the “Standard Normal Table”) in back of the text (or back of the supplement). Look up the closest standardized score in the table. Find the percentile corresponding to the standardized score (this is the percent of values below the corresponding standardized score or z-value). Chapter 13 18 Table B Standard Standard Standard Score Percentile Score Percentile Score Percentile –3.4 0.03 –1.1 13.57 1.2 88.49 –3.3 0.05 –1.0 15.87 1.3 90.32 –3.2 0.07 –0.9 18.41 1.4 91.92 –3.1 0.10 –0.8 21.19 1.5 93.32 –3.0 0.13 –0.7 24.20 1.6 94.52 –2.9 0.19 –0.6 27.42 1.7 95.54 –2.8 0.26 –0.5 30.85 1.8 96.41 –2.7 0.35 –0.4 34.46 1.9 97.13 –2.6 0.47 –0.3 38.21 2.0 97.73 –2.5 0.62 –0.2 42.07 2.1 98.21 –2.4 0.82 –0.1 46.02 2.2 98.61 –2.3 1.07 0.0 50.00 2.3 98.93 –2.2 1.39 0.1 53.98 2.4 99.18 –2.1 1.79 0.2 57.93 2.5 99.38 –2.0 2.27 0.3 61.79 2.6 99.53 –1.9 2.87 0.4 65.54 2.7 99.65 –1.8 3.59 0.5 69.15 2.8 99.74 –1.7 4.46 0.6 72.58 2.9 99.81 –1.6 5.48 0.7 75.80 3.0 99.87 –1.5 6.68 0.8 78.81 3.1 99.90 –1.4 8.08 0.9 81.59 3.2 99.93 –1.3 9.68 1.0 84.13 3.3 99.95 –1.2 11.51 1.1 86.43 3.4 99.97 Chapter 13 19 Table B: Percentiles of the Standardized Normal Distribution Standard Score Percentile (Z) -0.8 21.19 -0.7 24.20 -0.6 27.42 Chapter 13 20 Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of men are less than 68 inches tall? 24.20% 68 70 (height values) -0.71 0 (standardized values) Chapter 13 21 Health and Nutrition Examination Study of 1976-1980 (HANES) What height value is the 10th percentile for men ages 18 to 24? 10% ? 70 (height values) Chapter 13 22 Table B: Percentiles of the Standardized Normal Distribution SeeTable B (the “Standard Normal Table”) in back of the text (or back of the supplement). Look up the closest percentile in the table. Find the corresponding standardized score. Thevalue you seek is that many standard deviations from the mean. Chapter 13 23 Table B: Percentiles of the Standardized Normal Distribution Standard Score Percentile (Z) -1.4 8.08 -1.3 9.68 -1.2 11.51 Chapter 13 24 Health and Nutrition Examination Study of 1976-1980 (HANES) What height value is the 10th percentile for men ages 18 to 24? 10% ? 70 (height values) -1.3 0 (standardized values) Chapter 13 25 Observed Value for a Standardized Score What height value is the 10th percentile for men ages 18 to 24? observed value = mean plus [(standardized score) (std dev)] = 70 + [(-1.3 ) (2.8)] = 70 + (-3.64) = 66.36 Thevalue 66.36 is approximately the 10th percentile of the population. Chapter 13 26 Observed Value for a Standardized Score observed value = mean plus [(standardized score) (std dev)] x is the observed value m is the population mean z is the standardized score s is the population standard deviation x m zs Chapter 13 27 Health and Nutrition Examination Study of 1976-1980 (HANES) RECALL: What proportion of men are less than 68 inches tall? NOW: what proportion of men are greater than 68 inches tall? ? 100%-24.2% 24.20% = 75.8% 68 70 (height values) -0.71 0 (standardized values) Chapter 13 28 Health and Nutrition Examination Study of 1976-1980 (HANES) The average height of males ages 18– 24 years old was 70.0 inches with a standard deviation of 2.8 inches. Itis also known that this distribution of heights follows a normal or bell-shaped curve. What proportion of men are between 68 inches tall and 74 inches tall? Chapter 13 29 Health and Nutrition Examination Study of 1976-1980 (HANES) First, draw and label a normal curve. m = 70 in. s = 2.8 in. standard deviations: -3 -2 -1 0 1 2 3 height values: 61.6 64.4 67.2 70 72.8 75.6 78.4 Chapter 13 30 Health and Nutrition Examination Study of 1976-1980 (HANES) Shade on the graph the range of heights between 68 and 74 inches. ? (height values) 68 70 74 (standardized values) ? 0 ? Chapter 13 31 Standardized Scores How many standard deviations is 68 from 70? standardized 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 13 32 Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of men are less than 68 inches tall? ?% (height values) 68 70 74 (standardized values) -0.71 0 ? Chapter 13 33 Table B: Percentiles of the Standardized Normal Distribution Standard Score Percentile (Z) -0.8 21.19 -0.7 24.20 -0.6 27.42 Chapter 13 34 Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of men are less than 68 inches tall? 24.20% (height values) 68 70 74 (standardized values) -0.71 0 ? Chapter 13 35 Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of men are less than 74 inches tall? ?% 24.20% (height values) 68 70 74 (standardized values) -0.71 0 ? Chapter 13 36 Standardized Scores How many standard deviations is 74 from 70? standardized score = (observed value minus mean) / (std dev) [ = (74 - 70) / 2.8 = 1.43 ] The value 74 is 1.43 standard deviations above the mean 70. Chapter 13 37 Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of men are less than 74 inches tall? ?% 24.20% (height values) 68 70 74 (standardized values) -0.71 0 1.43 Chapter 13 38 Table B: Percentiles of the Standardized Normal Distribution Standard Score Percentile (Z) 1.3 90.32 1.4 91.92 1.5 93.32 Chapter 13 39 Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of men are less than 74 inches tall? 91.92% 24.20% (height values) 68 70 74 (standardized values) -0.71 0 1.43 Chapter 13 40 Health and Nutrition Examination Study of 1976-1980 (HANES) What proportion of men are between 68 inches tall and 74 inches tall? 67.72% 91.92% 91.92% – 24.20% 24.20% = 67.72% (height values) 68 70 74 (standardized values) -0.71 0 1.43 Chapter 13 41 Key Concepts Population values are distributed with differing shapes, some normal, some non-normal. Empirical Rule (“68-95-99.7 Rule”) Standardized Score Percentile Standard Normal Table Chapter 13 42

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posted: | 12/1/2009 |

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