Slide 1Slide 2Tallest Living HumansSlide 4The Standard Deviation as a RulerStandardizing with z-scoresStandardizing with z-scoresStandardizing with z-scores (cont.)Slide 9Units of Measure and z-ScoresDifferent Units of Measure and z-ScoresSlide 12When Is a z-score BIG?When Is a z-score Big? (cont.)Slide 15Slide 16When Is a z-score Big? (cont.)When is a z-score Big? (cont.)The 68-95-99.7 RuleThe 68-95-99.7 Rule (cont.)In-Class Activity – Groups of 2 (or 3)In-Class Activity (cont.)So, When Is a z-score Big?Working with Normal ModelsSlide 25Beyond the 68-95-99.7 RuleFinding Normal Percentages Using TechnologyFinding Normal Percentages Using TechnologyFinding Normal Percentages Using Technology (cont.)Verifying the 68-95-99.7 RuleIn-Class Activity – Groups of 2 (or 3)In-Class Activity (cont.)In-Class Activity (cont.)In-Class Activity (cont.)In-Class Activity (cont.)Given an Area, What is y?In-Class Activity (cont.)Slide 38Slide 39Are You Normal? Normal Probability Plots (cont)Are You Normal? Normal Probability Plots (cont)Slide 42How Straight is “Straight Enough”?Goodness of Fit TestGoodness of Fit Test (Cont.)Goodness of Fit Test (Cont.)WarningChapter Example #1Chapter Example #1, continuedChapter Example #1, continued1Chapter05 Presentation 0115Copyright © 2014, 2012, 2009 Pearson Education, Inc.Chapter 5The Standard Deviation as a Ruler and the Normal Model Most important2Chapter05 Presentation 0115Copyright © 2014, 2012, 2009 Pearson Education, Inc.5.1Standardizing with z-ScoresChapter05 Presentation 01153Copyright © 2014, 2012, 2009 Pearson Education, Inc.Tallest Living HumansRecent research estimates that the average height of men is 5’ 9” (69 inches) with a standard deviation of 3.0 inches. For women, the average is 5’ 3.5” (63.5 inches) with a standard deviation of 2.5 inches.Chapter05 Presentation 01154Copyright © 2014, 2012, 2009 Pearson Education, Inc.According to 2011 Edition of Guiness World Records, the tallest living man is Sultan Kösen of Turkey. He stands 8’ 3.0” (99.0 inches). The tallest living woman is Yao Defen of China. She stands 7’ 8.0” (92.0 inches).Sultan is taller than Yao, but is his height more unusual?Chapter05 Presentation 01155Copyright © 2014, 2012, 2009 Pearson Education, Inc.The Standard Deviation as a RulerThe standard deviation is the most common measure of variation.The trick in comparing very different-looking values is to use standard deviations as our rulers.Chapter05 Presentation 01156Copyright © 2014, 2012, 2009 Pearson Education, Inc.Standardizing with z-scoresWe compare individual data values to their mean, relative to their standard deviation using the following formula:z tells us how many standard deviations the value y is away from the mean. z y y sChapter05 Presentation 01157Copyright © 2014, 2012, 2009 Pearson Education, Inc.Standardizing with z-scoresChapter05 Presentation 01158Copyright © 2014, 2012, 2009 Pearson Education, Inc.Standardizing with z-scores (cont.)Calculate z-scores for Sultan (y=99.0”) and Yao (y=92.0”). Recall, for our Height example:Men WomenMean 69” 63.5”Std. Dev.3” 2.5”9Chapter05 Presentation 0115Copyright © 2014, 2012, 2009 Pearson Education, Inc.5.2Shifting and ScalingChapter05 Presentation 011510Copyright © 2014, 2012, 2009 Pearson Education, Inc.Units of Measure and z-ScoresIf you convert an entire data set into z-scores:This is known as standardizing the data.The z-scores are unitless numbers.This shifts the mean to 0, andThis rescales the standard deviation to 1.Shifting and rescaling a data set does not change the shape of the distribution.This applies to converting data into z-scores, or converting data from one unit of measure to another.Chapter05 Presentation 011511Copyright © 2014, 2012, 2009 Pearson Education, Inc.Different Units of Measure and z-ScoresDaily high temperature in Knoxville, TN from 05/01/08 to 05/31/08 in Fahrenheit (F°) and Celsius (C°).Calculate the Z-scores for the maximum values:C° = 5(F° - 32)9Z = Z = (87-77.22)/5.879= 1.6767 1.676712Chapter05 Presentation 0115Copyright © 2014, 2012, 2009 Pearson Education, Inc.5.3Normal ModelsChapter05 Presentation 011513Copyright © 2014, 2012, 2009 Pearson Education, Inc.When Is a z-score BIG?A z-score gives us an indication of how unusual a value is because it tells us how far it is from the mean.The z-scores for Sultan and Yao are at least 10, but these are world records (very unusual)! Where is the cutoff between typical and unusual values?Chapter05 Presentation 011514Copyright © 2014, 2012, 2009 Pearson Education, Inc.When Is a z-score Big? (cont.)There is no universal standard for z-scores, but there is a model that shows up over and over in Statistics.This model is called the Normal Model.Normal models are appropriate for distributions whose shapes are unimodal and symmetric.Chapter05 Presentation 011515Copyright © 2014, 2012, 2009 Pearson Education, Inc.This is a histogram of the heights (in inches) of 1500 women (mean 63.5”, standard deviation 2.5”) with a Normal model drawn on top of it.Chapter05 Presentation 011516Copyright © 2014, 2012, 2009 Pearson Education, Inc.This is the same histogram, showing where Yao Defen’s height falls (recall, her height was z=11.4 standard deviations above the mean). 92.0”Chapter05 Presentation 011517Copyright © 2014, 2012, 2009 Pearson Education, Inc.When Is a z-score Big? (cont.)There is a Normal model for every possible combination of mean and standard deviation. We write N(μ,σ) to represent a Normal model with a mean of μ and a standard deviation of σ.When we standardize Normal data, we still call the standardized value a z-score, and we write yzms-=Chapter05 Presentation 011518Copyright © 2014, 2012, 2009 Pearson Education, Inc.When is a z-score Big? (cont.)Once we have standardized, we need only one Normal model: The N(0,1) model is called the Standard Normal Model (or the Standard Normal Distribution).Chapter05 Presentation 011519Copyright © 2014, 2012, 2009 Pearson Education, Inc.The 68-95-99.7 RuleWe will be more precise in the near future, but until then we will use a simple rule that tells us a lot about the Normal model.On exams, be very familiar with this rule!Chapter05 Presentation 011520Copyright © 2014, 2012, 2009 Pearson Education, Inc.The 68-95-99.7 Rule (cont.)The following shows what the 68-95-99.7
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