UK STA 200 - Chapter 13 Normal Distributions

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Chapter 13Thought Question 1Thought Question 3Thought Question 4Bell-Shaped Curve: The Normal Distribution of Population ValuesAsymmetric Distributions of the Population ValuesThe Normal DistributionWith the Mean and Standard Deviation of the Normal Distribution We Can Determine:Empirical Rule for Any Normal CurveSlide 10Health and Nutrition Examination Study of 1976-1980 (HANES)Slide 12Slide 13Slide 14Standardized ScoresSlide 16Slide 17Table B: Percentiles of the Standardized Normal DistributionTable BSlide 20Slide 21Slide 22Slide 23Slide 24Slide 25Observed Value for a Standardized ScoreSlide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Key ConceptsChapter 13 1Chapter 13Normal DistributionsChapter 13 2 Thought Question 2What does it mean if a person’s SAT score falls at the 20th percentile for all people who took the test?Birth weights of babies born in the United States follow, at least approximately, a bell-shaped curve. What does that mean?Thought Question 1Chapter 13 3Thought Question 3A 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 4Thought Question 4Many 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 5Bell-Shaped Curve:The Normal Distributionof Population ValuesChapter 13 6Asymmetric Distributionsof the Population ValuesChapter 13 7The Normal Distributionstandard deviationmeanChapter 13 8With 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 percentileChapter 13 9Empirical Rule forAny 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 10Empirical Rule forAny Normal Curve+1sd-1sd68%+2 sd-2 sd95%+3 sd-3 sd99.7%Chapter 13 11Health 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 inchesChapter 13 12Health 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 inchesChapter 13 130 1-1 2-2 3-3standard deviations:70 72.867.2 75.664.4 78.461.6height values:Health and Nutrition Examination Study of 1976-1980(HANES)What proportion of men are less than 72.8 inches tall?? 70 72.8 (height values) +168%(by Empirical Rule)-1? = 84%16%32% / 2 =Chapter 13 14Health and Nutrition Examination Study of 1976-1980(HANES)What proportion of men are less than 68 inches tall??68 70 (height values)0 1-1 2-2 3-3standard deviations:70 72.867.2 75.664.4 78.461.6height values:Chapter 13 15Standardized 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 16Standardized Scoresstandardized score =(observed value minus mean) / (std dev)-z is the standardized score-x is the observed value- is the population mean- is the population standard deviationzx=−μσChapter 13 17Health and Nutrition Examination Study of 1976-1980(HANES)What proportion of men are less than 68 inches tall?-0.71 0 (standardized values)68 70 (height values)?Chapter 13 18Table 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 19Table BStandard Standard StandardScore 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 99.97Chapter 13 20Table B: Percentiles of the Standardized Normal DistributionStandard Score(Z)Percentile-0.8 21.19-0.7 24.20-0.6 27.42Chapter 13 21-0.71 0 (standardized values)68 70 (height values)Health and Nutrition Examination Study of 1976-1980(HANES)What proportion of men are less than 68 inches tall?24.20%Chapter 13 22Health 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 23Table 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 percentile in the


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UK STA 200 - Chapter 13 Normal Distributions

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