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Warm-up We are going to collect some data and determine if it is “normal” Roll each pair of dice 10 times and record the SUM of the two numbers in your table Plot the 10 sums on the dotplot on the board Describe the distribution of sums Slide 6 -1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Unit 2 Lesson 2: The Normal Distribution At the end of today’s lesson, you will know… Objectives: 1) Apply the Empirical 68-95-99.7 Rule 2.) Find normal proportions, given mean and standard deviation. 3.) Use a normal probability plot to determine if the normal model is reasonable Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 2 Part I WHAT ABOUT THAT NORMAL CURVE? Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 3 Normal Curves • Special density curves that are symmetric, singlepeaked and bell-shaped • Distributions they describe are called normal distributions • Completely described by its mean and standard deviation • This is abbreviated by N(μ, σ) • If you change the mean (center) but not the standard deviation (spread), it moves the normal curve • If you make the standard deviation larger, it spreads out the curve. If you make the standard deviation smaller, it squeezes the curve. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Normal Curves Why study Normal Curves? Some reasons we are interested in normal distributions are: Normal distributions are good descriptions for some distributions of real data. Normal distributions are good approximations to the results of many kinds of chance outcomes. Many statistical inference procedures based on normal distributions work well for other roughly symmetrical distributions. A variety of psychological test scores and physical phenomena like photon counts can be well approximated by a normal distribution. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Part II 68-96-99.7 (EMPIRICAL) RULE Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 6 II. 68-95-99.7 Rule – aka “Empirical Rule” Normal models give us an idea of how extreme a value is by telling us how likely it is to find one that far from the mean. We can find these numbers precisely, but until then we will use a simple rule that tells us a lot about the Normal model… Slide 6 -7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. The 68-95-99.7 Rule (cont.) It turns out that in a Normal model: about 68% of the values fall within one standard deviation of the mean; about 95% of the values fall within two standard deviations of the mean; and, about 99.7% (almost all!) of the values fall within three standard deviations of the mean. Slide 6 -8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. The 68-95-99.7 Rule (cont.) The following shows what the 68-95-99.7 Rule tells us: Slide 6 -9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Empirical Rule (68 – 95 – 99.7 Rule) Suppose IQ scores of students at IM Smart College follow a normal distribution with µ = 115 and σ = 12. a.) Draw and label the normal curve: b.) What % of students scored between 103 and 127? c.) What % scored between 115 and 127? d.) What % scored higher than 139? e.) What % scored below 103? f.) A score of 127 corresponds to what percentile? Copyright © 2010, 2007, 2004 Pearson Education, Inc. Part III STANDARD NORMAL DISTRIBUTION Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 11 Standard Normal Distribution In standard normal distribution, the mean = 0 and the standard deviation = 1. Standardized variable (z) is found: z Copyright © 2010, 2007, 2004 Pearson Education, Inc. x Slide 6 - 12 But why use the standard? Remember that the area under a density curve is a proportion of the observations in a distribution. In a standard Normal distribution, the 68-95-99.7 rule tell us about 68% fall between z = 1 and z = – 1. and 95% fall between z = 2 and z = –2, etc. What if we want to find the percent of observations that fall between z = -1.25 and z = 1.25? (the 68-9599.7 rule can’t help us) Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 13 So let’s standardize! Standard Normal Calculations Refer to your Statistics Packet and find Table A Table A is a table of areas under the standard Normal curve. How you read the table: The left bold column represents the z value to the tenth. The top bold column represents the hundredth of the z value The table entry for each value of z is the area under the curve to the left of z. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 14 Standard Normal Calculations a) Find the proportion of observations from the standard normal distribution that have a z value that is LESS THAN 2.34. b) Find the proportion of observations from the standard normal distribution that have a z value that is GREATER THAN -1.5. c) Find the proportion of observations from the standard normal distribution that have a z value that is GREATER THAN 0. d.) What z-values will give you 25% of the observations? Copyright © 2010, 2007, 2004 Pearson Education, Inc. Part IV ARE YOU NORMAL? HOW CAN YOU TELL? Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 16 IV. Are You Normal? How Can You Tell? When you actually have your own data, you must check to see whether a Normal model is reasonable. We have done this two ways: 1) Looking at a histogram to determine if data is roughly unimodal and symmetric. 2) Calculate the mean and standard deviation and determine the percentage of values that fall within one, two, and three standard deviations and check against the 68-95-99.7 Rule. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 17 Are You Normal? How Can You Tell? (cont.) A more specialized graphical display that can help you decide whether a Normal model is appropriate is the Normal probability plot. If the distribution of the data is roughly Normal, the Normal probability plot approximates a diagonal straight line. Deviations from a straight line indicate that the distribution is not Normal. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 18 Normal Probability Plots A normal probability plot takes each data value and plots it against the z-score you would expect that value to have if the data were perfectly normal. Since Normal distributions are symmetric, the data should appear as a straight line. We will use the calculator to create normal probability plots but here are some examples first. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 19 Nearly Normal data have a histogram and a Normal probability plot that look somewhat like this example: Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 20 A skewed distribution might have a histogram and Normal probability plot like this: Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 21 Example: Fourth grade boys were given an agility test. They jumped from side to side across a set of parallel lines, counting the number of lines they clear in 30 seconds. Here are their scores: 22, 17, 18, 29, 22, 23, 24, 23, 17, 21 Enter the data into L1. Turn StatPlot On and choose the last graph icon. Your data axis can be X or Y. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 6 - 22