The Normal DistributionThe “68-95-99.7 Rule”The Empirical RuleThe Standard Normal DistributionSlide 5Areas Under the Standard Normal CurveSlide 7Slide 8Slide 9Slide 10Slide 11Slide 12TI-83 – Standard Normal AreasStandard Normal AreasOther Normal CurvesTI-83 – Area Under Normal CurvesIQ ScoresSlide 18The Standard Normal TableSlide 20Slide 21Slide 22The Three Basic ProblemsSlide 24Tables – Area Under Normal CurvesSlide 26Slide 27Slide 28Slide 29Z-ScoresAreas Under Other Normal CurvesExampleBag A vs. Bag BSlide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48The Normal DistributionLecture 22Section 6.3.1Tue, Oct 2, 2007The “68-95-99.7 Rule”For any normal distribution, Approximately 68% of the values lie within one standard deviation of the mean.Approximately 95% of the values lie within two standard deviations of the mean.Approximately 99.7% of the values lie within three standard deviations of the mean.The Empirical RuleThe well-known Empirical Rule is similar, but more general.If a distribution has a “mound shape”, thenApproximately 68% lie within one standard deviation of the mean.Approximately 95% lie within two standard deviations of the mean.Nearly all lie within three standard deviations of the mean.The Standard Normal DistributionThe standard normal distribution It is denoted by the letter Z.That is, Z is N(0, 1).The Standard Normal Distribution0 1 2 3-1-2-3N(0, 1)zAreas Under the Standard Normal CurveEasy questions:What is the total area under the curve?What proportion of values of Z will fall below 0?What proportion of values of Z will fall above 0?Areas Under the Standard Normal CurveHarder questions:What proportion of values will fall below +1?What proportion of values will fall above +1?What proportion of values will fall below –1?What proportion of values will fall between –1 and +1?Areas Under the Standard Normal CurveIt turns out that the area to the left of +1 is 0.8413.0 1 2 3-1-2-30.8413zAreas Under the Standard Normal CurveSo, what is the area to the right of +1?0 1 2 3-1-2-30.8413Area?zAreas Under the Standard Normal CurveSo, what is the area to the left of -1?0 1 2 3-1-2-30.8413Area?zAreas Under the Standard Normal CurveSo, what is the area between -1 and 1?0 1 2 3-1-2-30.8413Area?0.8413zAreas Under the Standard Normal CurveThere are two methods to finding standard normal areas:The TI-83 function normalcdf.Standard normal table.I will show you how to use the table, but we will normally use the TI-83.TI-83 – Standard Normal AreasPress 2nd DISTR.Select normalcdf (Item #2).Enter the lower and upper bounds of the interval.If the interval is infinite to the left, enter -E99 as the lower bound.If the interval is infinite to the right, enter E99 as the upper bound.Press ENTER.Standard Normal AreasUse the TI-83 to find the following.The area between -1 and +1.The area to the left of +1.The area to the right of +1.Other Normal CurvesIf we are working with a different normal distribution, say N (30, 5), then how can we find areas under the curve?TI-83 – Area Under Normal CurvesUse the same procedure as before, except enter the mean and standard deviation as the 3rd and 4th parameters of the normalcdf function.Find area between 25 and 38 in the distribution N(30, 5).IQ ScoresIntelligence Quotient.Understanding and Interpreting IQ.IQ scores are standardized to have a mean of 100 and a standard deviation of 15.Psychologists often assume a normal distribution of IQ scores as well.IQ ScoresWhat percentage of the population has an IQ above 120? above 140?What percentage of the population has an IQ between 75 and 125?The Standard Normal TableSee pages 406 – 407 or pages A-4 and A-5 in Appendix A.The table is designed for the standard normal distribution.The entries in the table are the areas to the left of the z-value.The Standard Normal TableTo find the area to the left of +1, locate 1.00 in the table and read the entry.z .00 .01 .02 …: : : : …0.9 0.8159 0.8186 0.8212 …1.0 0.8413 0.8438 0.8461 …1.1 0.8643 0.8665 0.8686 …: : : : …The Standard Normal TableTo find the area to the left of 2.31, locate 2.31 in the table and read the entry.z .00 .01 .02 …: : : : …2.2 0.9861 0.9864 0.9868 …2.3 0.9893 0.9896 0.9898 …2.4 0.9918 0.9920 0.9922 …: : : : …The Standard Normal TableThe area to the left of 1.00 is 0.8413.That means that 84.13% of the population is below 1.00.0 1 2 3-1-2-30.8413The Three Basic ProblemsaabaFind the area to the left of a:Look up the value for a.Find the area to the right of a:Look up the value for a; subtract it from 1.Find the area between a and b:Look up the values for a and b; subtract the smaller value from the larger.Standard Normal AreasUse the Standard Normal Tables to find the following.The area between -2.14 and +1.36.The area to the left of -1.42.The area to the right of -1.42.Tables – Area Under Normal CurvesIf X is N (30, 5), what is the area to the left of 35?30 35 40 45252015Tables – Area Under Normal CurvesIf X is N (30, 5), what is the area to the left of 35?30 35 40 45252015Tables – Area Under Normal CurvesIf X is N (30, 5), what is the area to the left of 35?30 35 40 45252015?Tables – Area Under Normal CurvesIf X is N (30, 5), what is the area to the left of 35?30 35 40 452520150 1 2 3-1-2-3XZ?Tables – Area Under Normal CurvesIf X is N (30, 5), what is the area to the left of 35?30 35 40 452520150.84130 1 2 3-1-2-3XZZ-ScoresZ-score, or standard scoreCompute the z-score of x asorEquivalentlyorsxxzzsxx zx xzAreas Under Other Normal CurvesIf a variable X has a normal distribution, then the z-scores of X have a standard normal distribution.)1,0( is then ,),( is If NXNXExampleLet X be N(30, 5).What proportion of values of X are below 38?Compute z = (38 – 30)/5 = 8/5 = 1.6.Find the area to the left of 1.6 under the standard normal curve.Answer: 0.9452.Therefore, 94.52% of the values of X are below 38.Bag A vs. Bag BSuppose we have two bags, Bag A and Bag B.Each bag contains millions of vouchers.In Bag A, the values of the vouchers have distribution N(50, 10). In Bag B, the values of the vouchers have distribution N(80,
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