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Bell-Shaped Curves and Other Shapes Thought Questions 1. The heights of adult women in the United States follow, at least approximately, a bell-shaped curve. What do you think this means? 2. What does it mean to say that a man’s weight is in the 30th percentile for all adult males? 3. A “standardized score” is simply the number of standard deviations an individual falls above or below the mean for the whole group. Male heights have a mean of 70 inches and a standard deviation of 3 inches. Female heights have a mean of 65 inches and a standard deviation of 2 ½ inches. Thus, a man who is 73 inches tall has a standardized score of 1. What is the standardized score corresponding to your own height? Bell-Shaped Curves and Other Shapes Frequency Curves Smoothed-out histogram by connecting tops of rectangles with smooth curve. Frequency curve for population of British male heights. The measurements follow a normal distribution (or a bell-shaped or Gaussian curve). Height of curve set so area under entire curve is 1. Frequency curve for population of dollar amounts of car insurance damage claims. The measurements follow a right skewed distribution. Majority of claims were below $5,000, but there were occasionally a few extremely high claims. Bell-Shaped Curves and Other Shapes Proportions Key: Proportion of population of measurements falling in a certain range = area under curve over that range. Mean British Height is 68.25 inches. Area to the right of the mean is 0.50. So about half of all British men are 68.25 inches or taller. Many populations of measurements follow approximately a normal curve: • Physical measurements within a homogeneous population – heights of male adults. • Standard academic tests given to a large group – SAT scores. Bell-Shaped Curves and Other Shapes Percentiles and Standardized Scores Your percentile = the percentage of the population that falls below you. Finding percentiles for normal curves requires: • Your own value. • The mean for the population of values. • The standard deviation for the population. Then any bell curve can be standardized so one table can be used to find percentiles. Proportions and Percentiles for Standard Normal Scores Standard Score, z -6.00 -5.20 -4.26 -3.00 : -1.00 : -0.58 : 0.00 Proportion Below z 0.000000001 0.0000001 0.00001 0.0013 : 0.16 : 0.28 : 0.50 Percentile 0.0000001 0.00001 0.001 0.13 : 16 : 28 : 50 Standard Score, z 0.03 0.05 0.08 0.10 : 0.58 : 1.00 : 6.00 Proportion Below z 0.51 0.52 0.53 0.54 : 0.72 : 0.84 : 0.999999999 Percentile 51 52 53 54 : 72 : 84 : 99.9999999 Bell-Shaped Curves and Other Shapes Standard Normal Curve Standard Normal Curve 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 3.4 3.1 2.8 2.5 2.2 1.9 1.6 1.3 1 0.7 0.4 0.1 -0.2 -0.5 -0.8 -1.1 -1.4 -1.7 -2 -2.3 -2.6 -2.9 -3.2 -3.5 0 The standard normal curve has a mean of 0 and standard deviation of 1. We convert our data values to the their corresponding values on the standard normal curve. This enables us to get exact probabilities of certain events. Standardized Score (or z-score) is : observed value – mean standard deviation Bell-Shaped Curves and Other Shapes Empirical Rule For any normal curve, approximately … 68% of the values fall within 1 standard deviation of the mean in either direction 95% of the values fall within 2 standard deviations of the mean in either direction 99.7% of the values fall within 3 standard deviations of the mean in either direction Example Some IQ tests are standardized to a Normal Model, with a mean of 100 and a standard deviation of 16 Approximately 95% of the IQ scores are expected to be within the interval 68 to 132 IQ points. Approximately 16% of IQ scores are expected to be above 116 IQ points. Approximately 2.5% of the IQ scores are expected to be above 132. Bell-Shaped Curves and Other Shapes Heights of Adult Women in US Heights of US women in US 0.18 Height of curve 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 55 55.7 56.4 57.1 57.8 58.5 59.2 59.9 60.6 61.3 62 62.7 63.4 64.1 64.8 65.5 66.2 66.9 67.6 68.3 69 69.7 70.4 71.1 71.8 72.5 73.2 73.9 74.6 0 Heights (in inches) Adult women in U.S. have a mean height of 65 inches with a standard deviation of 2.5 inches and heights are approximately bell-shaped. 68% of adult women are between 62.5 and 67.5 inches, 95% of adult women are between 60 and 70 inches, 99.7% of adult women are between 57.5 and 72.5 inches. Bell-Shaped Curves and Other Shapes Standardized Scores (or z-scores) Standardized Score (or z-score) is Heights of US women observed value – mean standard deviation 0.18 Height of curve 0.16 0.14 0.12 0.1 Remaining Area 0.08 Area less 61 inches 0.06 0.04 Height of US women - Population Mean height = 65 Standard deviation = 2.5 What is the probability that a woman selected at random from this population will be less than 61 inches? 0.02 55 56.4 57.8 59.2 60.6 62 63.4 64.8 66.2 67.6 69 70.4 71.8 73.2 74.6 0 Probability(height is less than 61inches)? Standard Normal Curve z-score = (61 – 65)/2.5 = -4/2.5 = -1.6 0.45 0.4 61 is 1.6 standardized scores from the mean of 65. 0.35 0.25 Remaing area 0.2 From Tables 8.1 (pg 157 in text): Area less than -1.6 0.15 Prob(z-score less than -1.6) = .05 0.1 0.05 0 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Axis Title 0.3 The probability that a women selected at random will have a height less than 61 inches is 0.05. Bell-Shaped Curves and Other Shapes Example: Cholesterol Assume the cholesterol levels of adult American women can be described by a Normal model with a mean of 188 mg/dL and a standard deviation of 24. What is the probability that a randomly selected adult women will have a cholesteral level between 150 and 170 Mg/dL? Cholesterol Level of 150: Standardized score(z-score) = 150 – 188/24 = -1.583 Cholesterol Level of 170: Standardized score(z-score) = 170 – 188/24 = -0.75 Probability(Cholesterol level between 150 and 170) = Probability(z-score is between -1.583 and -0.75) From Table 8.1 (rounded to nearest decimal): Proportion below z=-0.74 minus proportion below z=-1.55 => (0.23 – 0.06) = 0.17 The probability that a randomly selected adult women will have a cholesteral level between 150 and 170 Mg/dL is 0.17. Bell-Shaped Curves and Other Shapes Using Excel to Find Normal Curve Proportions Example: Heights of Adult Women in US mean height = 65 standard deviation = 2.5. What is the probability that a woman selected at random from this population will be less than 61 inches? Excel: In function line type =NORMDIST(height value, mean, standard deviation, 1) For this example, =NORMDIST(61, 65,2.5,1) equals 0.055 Can also use =NORMSDIST(z-score value). For this example, z-score = (61 – 65)/2.5 = -4/2.5 = -1.6 So entering = NORMSDIST(-1.6) also equals 0.05 Example: Cholesterol mean = 188 standard deviation =24 Probability(Cholesterol level between 150 and 170) = NORMDIST(170,188,24,1) – NORMDIST(150,188,24,1) equals 0.169954598 Probability(z-score is between -1.583 and -0.75) =NORMSDIST(-0.75) - NORMSDIST(-1.583) equals 0.16991662 Bell-Shaped Curves and Other Shapes Journal’s Paper on ESP Expected to Prompt Outrage – NY Times – January 5th 2011 •One of psychology’s most respected journals has agreed to publish a paper presenting what its author describes as strong evidence for extrasensory perception, the ability to sense future events. •The paper describes nine unusual lab experiments performed over the past decade by its author, Daryl J. Bem, an emeritus professor at Cornell, testing the ability of college students to accurately sense random events, like whether a computer program will flash a photograph on the left or right side of its screen. •“It’s craziness, pure craziness. I can’t believe a major journal is allowing this work in,” Ray Hyman, an emeritus professor of psychology at the University Oregon and longtime critic of ESP research, said. “I think it’s just an embarrassment for the entire field.” •The editor of the journal, Charles Judd, a psychologist at the University of Colorado, said the paper went through the journal’s regular review process. “Four reviewers made comments on the manuscript,” he said, “and these are very trusted people.” •All four decided that the paper met the journal’s editorial standards, Dr. Judd added, even though “there was no mechanism by which we could understand the results.” Bell-Shaped Curves and Other Shapes Study: Experimental Evidence for Anomalous Retroactive Influences on Cognition and Affect •The term psi denotes anomalous processes of information or energy transfer that are currently unexplained in terms of known physical or biological mechanisms. •This article reports nine experiments designed to test for such retroactive influence by “timereversing” several well-established psychological effects so that the individual’s responses are obtained before the putatively causal stimulus events occur. •The trend is exemplified by several recent “presentiment” experiments, pioneered by Radin (1997), in which physiological indices of participants’ emotional arousal were monitored as participants viewed a series of pictures on a computer screen. •Most of the pictures were emotionally neutral, but a highly arousing negative or erotic image was displayed on randomly selected trials. •As expected, strong emotional arousal occurred when these images appeared on the screen, but the remarkable finding is that the increased arousal was observed to occur a few seconds before the picture appeared, before the computer has even selected the picture to be displayed. Bell-Shaped Curves and Other Shapes Study: Experimental Evidence for Anomalous Retroactive Influences on Cognition and Affect Experiment 1: Precognitive Detection of Erotic Stimuli One hundred Cornell undergraduates, 50 women and 50 men, were recruited for this experiment using the Psychology Department’s automated online sign-up system. Methods: On each trial of the experiment, pictures of two curtains will appear on the screen side by side. One of them has a picture behind it; the other has a blank wall behind it. •Your task is to click on the curtain that you feel has the picture behind it. The curtain will then open, permitting you to see if you selected the correct curtain. There will be 36 trials in all. •Several of the pictures contain explicit erotic images (e.g., couples engaged in nonviolent but explicit consensual sexual acts). •If you object to seeing such images, you should not participate in this experiment. Results: Across all 100 sessions, participants correctly identified the future position of the erotic pictures significantly more frequently than the 50% hit rate expected by chance: 53.1%, t(99) = 2.51, p = .01 In contrast, their hit rate on the nonerotic pictures did not differ significantly from chance: 49.8%, t(99) = -0.15, p = .56 Bell-Shaped Curves and Other Shapes Text Questions