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Normal Probability Distributions Chapter 5 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Addison Wesley Longman 1 Chapter 5 Normal Probability Distributions 5-1 Overview 5-2 The Standard Normal Distribution 5-3 & 5-4 Nonstandard Normal Distributions (Finding Probabilities & Finding Scores) 5-5 The Central Limit Theorem 5-6 Normal Distributions as Approximation to Binomial Distribution Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 5-2 Distributions of continuous random variables The Uniform Distribution & The Standard Normal Distribution Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 Uniform Distribution Definition a probability distribution in which every value of the random variable is equally likely – The probability distribution is call a uniform distribution – The random variable is called a random variable with a uniform distribution Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 Definition Density Curve a graph of a continuous probability distribution that satisfies the following two properties: 1. the total area under the curve must be 1 2. every point on the curve must have a vertical height that is 0 or greater A density curve is the graph of a continuous random variable, so the area under the curve is 1 and there is a correspondence between area and probability. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 Uniform Distributions Shaded Area = L • W = 3 • 0.2 = 0.6 P( 1º < x < 4º ) = 0.6 P(x ) P( x > 1º ) = 0.8 P(x ) 3 0.2 0.2 x 0 0 1 2 3 4 5 Temperature (degrees Celsius) x 0 0 1 2 3 4 5 Temperature (degrees Celsius) Figure 5-3 Figure 5-2 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 Different example: Heights of Adult Men and Women Women: µ = 63.6 s = 2.5 Men: µ = 69.0 s = 2.8 63.6 69.0 Height (inches) Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 Normal Distribution Continuous random variable Normal distribution Curve is bell shaped and symmetric µ Score Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 Normal Distribution Continuous random variable Normal distribution Curve is bell shaped and symmetric µ Score Formula 5-1 y= e Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 2 s ( x–µ 2 s ) 2 p 9 Definition Standard Normal Distribution a normal probability distribution that has a mean of 0 and a standard deviation of 1 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10 Definition Standard Normal Distribution a normal probability distribution that has a mean of 0 and a standard deviation of 1 •Area found in •Table A-2 •Area = 0.3413 •0.4429 -3 -2 -1 0 1 Score (z ) 2 3 Figure 5-5 0 z = 1.58 Figure 5-6 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 Standard Normal Distribution s=1 µ=0 ? 0 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman x 12 Table A-2 Back left cover of text book Formula card Appendix Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 Table A-2 Standard Normal Distribution s=1 µ=0 0 x z Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 Table A-2 Standard Normal (z) Distribution z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 .0000 .0398 .0793 .1179 .1554 .1915 .2257 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 .0040 .0438 .0832 .1217 .1591 .1950 .2291 .2611 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2642 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2995 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .0319 .0714 .1103 .1480 .1844 .2190 .2517 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .0359 .0753 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 To find: z Score the distance along horizontal scale on graph; refer to the leftmost column (and top row) in Table A-2 Area (or probability) the region under the curve; refer to the numbers in the body of Table A-2 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16 Table A-2 Standard Normal Distribution s=1 µ=0 .4495 0 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman z = 1.64 17 Table A-2 Standard Normal (z) Distribution z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 .0000 .0398 .0793 .1179 .1554 .1915 .2257 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 .0040 .0438 .0832 .1217 .1591 .1950 .2291 .2611 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2642 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2995 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .0319 .0714 .1103 .1480 .1844 .2190 .2517 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .0359 .0753 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 * Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18 Table A-2 Standard Normal Distribution s=1 µ=0 .4949 0 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman z = 2.57 19 Table A-2 Standard Normal (z) Distribution z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 .0000 .0398 .0793 .1179 .1554 .1915 .2257 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 .0040 .0438 .0832 .1217 .1591 .1950 .2291 .2611 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2642 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2995 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .0319 .0714 .1103 .1480 .1844 .2190 .2517 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .0359 .0753 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman * 20 Using Symmetry to Find the Area to the Left of the Mean Because of symmetry, these areas are equal. (a) (b) 0.4925 0.4925 0 z = –2.43 0 Equal distance away from 0 z = 2.43 • NOTE: Although a z score can be negative, the area under the curve (or the corresponding probabilities) can never be negative. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 21 Table A-2 Standard Normal (z) Distribution z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 .0000 .0398 .0793 .1179 .1554 .1915 .2257 .2580 .2881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .4452 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918 .4938 .4953 .4965 .4974 .4981 .4987 .0040 .0438 .0832 .1217 .1591 .1950 .2291 .2611 .2910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 .4940 .4955 .4966 .4975 .4982 .4987 .0080 .0478 .0871 .1255 .1628 .1985 .2324 .2642 .2939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922 .4941 .4956 .4967 .4976 .4982 .4987 .0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .4484 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925 .4943 .4957 .4968 .4977 .4983 .4988 .0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2995 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .4495 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931 .4948 .4961 .4971 .4979 .4985 .4989 .0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932 .4949 .4962 .4972 .4979 .4985 .4989 .0319 .0714 .1103 .1480 .1844 .2190 .2517 .2823 .3106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934 .4951 .4963 .4973 .4980 .4986 .4990 .0359 .0753 .1141 .1517 .1879 .2224 .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .4545 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936 .4952 .4964 .4974 .4981 .4986 .4990 * * * Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 22 The Empirical Rule Standard Normal Distribution: µ = 0 and s = 1 0 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 23 The Empirical Rule Standard Normal Distribution: µ = 0 and s = 1 68% within 1 standard deviation 0.340 –1 0.340 0 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 24 The Empirical Rule Standard Normal Distribution: µ = 0 and s = 1 95% within 2 standard deviations 68% within 1 standard deviation 0.340 0.340 0.135 –2 0.135 –1 0 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 2 25 The Empirical Rule Standard Normal Distribution: µ = 0 and s = 1 99.7% of data are within 3 standard deviations of the mean 95% within 2 standard deviations 68% within 1 standard deviation 0.340 0.340 0.024 0.024 0.001 0.001 0.135 –3 –2 0.135 –1 0 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 2 3 26 Probability of Half of a Distribution 0.5 0 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 27 Finding the Area to the Right of z = 1.27 Value found in Table A-2 0.3980 0 This area is 0.5 – 0.3980 = 0.1020 z = 1.27 Figure 5-8 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 28 Finding the Area Between z = 1.20 and z = 2.30 0.4893 (from Table A-2 with z = 2 .30) 0.3849 0 Area A is 0 .4893 – 0 .3849 = 0 .1044 A z = 1.20 z = 2.30 Figure 5-9 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 29 Notation P(a < z < b) denotes the probability that the z score is between a and b P(z > a) denotes the probability that the z score is greater than a P (z < a) denotes the probability that the z score is less than a Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 30 Figure 5-10 Interpreting Area Correctly Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 31 Figure 5-10 "greater than x" "at least x" "more than x" "not less than x" Interpreting Area Correctly Add to 0. 5 Subtract from 0. 5 0. 5 x x Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 32 Figure 5-10 "greater than x" "at least x" "more than x" "not less than x" Interpreting Area Correctly Add to 0. 5 Subtract from 0. 5 0. 5 x "less than x" "at most x" "no more than x" "not greater than x" x Subtract from 0.5 Add to 0. 5 0.5 x x Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 33 Figure 5-10 "greater than x" "at least x" "more than x" "not less than x" Interpreting Area Correctly Add to 0. 5 Subtract from 0. 5 0. 5 x "less than x" "at most x" "no more than x" "not greater than x" x Subtract from 0.5 Add to 0. 5 0.5 x x Add C Use A=C–B A B "between x1 and x2 " x1 x2 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman x1 x 2 34 Finding z Scores when Given Probabilities 95% 5% 5% or 0.05 0.50 0.45 0 FIGURE 5-11 z Finding the 95th Percentile Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 35 Finding z Scores when Given Probabilities 10% 90% Bottom 10% 0.10 0.40 z FIGURE 5-12 0 Finding the 10th Percentile Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 36 Finding a z - score when given a probability 1. Identify the probability representing an area bounded by the centerline. 2. Using the probability representing the area bounded by the centerline, locate the closest probability in the body of Table A-2 and identify the corresponding z score. 3. If the z score is positioned to the left of the centerline, make it a negative. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 37 Summary Two Distributions (continuous) Uniform distribution Standard Normal distribution Density Curve Finding Probabilities & Finding Scores Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 38