1. 1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Chapter 5 Normal Probability Distributions

2. 2 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman  Continuous random variable  Normal distribution Overview 5-1

3. 3 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman  Continuous random variable  Normal distribution Curve is bell shaped and symmetric µ Score Overview 5-1 Figure 5-1

4. 4 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman  Continuous random variable  Normal distribution Curve is bell shaped and symmetric µ Score Formula 5-1 Overview 5-1 Figure 5-1 x - µ 2 y = 1 2  e  2  ( )

5. 5 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5-2 The Standard Normal Distribution

6. 6 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman  Uniform Distribution a probability distribution in which the continuous random variable values are spread evenly over the range of possibilities; the graph results in a rectangular shape. Definitions

7. 7 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman  Density Curve (or probability density function) the graph of a continuous probability distribution Definitions

8. 8 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman  Density Curve (or probability density function) :  The graph of a continuous probability distribution Definitions 1. The total area under the curve must equal 1. 2. Every point on the curve must have a vertical height that is 0 or greater.

9. 9 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Because the total area under the density curve is equal to 1, there is a correspondence between area and probability.

10. 10 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Times in First or Last Half Hours Figure 5-3

11. 11 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Heights of Adult Men and Women Women: µ = 63.6  = 2.5 Men: µ = 69.0  = 2.8 69.0 63.6 Height (inches) Figure 5-4

12. 12 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition Standard Normal Deviation a normal probability distribution that has a mean of 0 and a standard deviation of 1

13. 13 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition Standard Normal Deviation a normal probability distribution that has a mean of 0 and a standard deviation of 1 0123-2-3 0 z = 1.58 Figure 5-5 Figure 5-6 Area = 0.3413 Area found in Table A-2 0.4429 Score (z )

14. 14 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Table A-2 Standard Normal Distribution µ = 0  = 1 0 x z

15. 15 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman.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 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 3.0.0000.0398.0793.1179.1554.1915.2257.2580.2881.3159.3413.3643.3849.4032.4192.4332.4452.4554.4641.4713.4772.4821.4861.4893.4918.4938.4953.4965.4974.4981.4987.0040.0438.0832.1217.1591.1950.2291.2611.2910.3186.3438.3665.3869.4049.4207.4345.4463.4564.4649.4719.4778.4826.4864.4896.4920.4940.4955.4966.4975.4982.4987.0080.0478.0871.1255.1628.1985.2324.2642.2939.3212.3461.3686.3888.4066.4222.4357.4474.4573.4656.4726.4783.4830.4868.4898.4922.4941.4956.4967.4976.4982.4987.0120.0517.0910.1293.1664.2019.2357.2673.2967.3238.3485.3708.3907.4082.4236.4370.4484.4582.4664.4732.4788.4834.4871.4901.4925.4943.4957.4968.4977.4983.4988.0160.0557.0948.1331.1700.2054.2389.2704.2995.3264.3508.3729.3925.4099.4251.4382.4495.4591.4671.4738.4793.4838.4875.4904.4927.4945.4959.4969.4977.4984.4988.0199.0596.0987.1368.1736.2088.2422.2734.3023.3289.3531.3749.3944.4115.4265.4394.4505.4599.4678.4744.4798.4842.4878.4906.4929.4946.4960.4970.4978.4984.4989.0279.0675.1064.1443.1808.2157.2486.2794.3078.3340.3577.3790.3980.4147.4292.4418.4525.4616.4693.4756.4808.4850.4884.4911.4932.4949.4962.4972.4979.4985.4989.0319.0714.1103.1480.1844.2190.2517.2823.3106.3365.3599.3810.3997.4162.4306.4429.4535.4625.4699.4761.4812.4854.4887.4913.4934.4951.4963.4973.4980.4986.4990.0359.0753.1141.1517.1879.2224.2549.2852.3133.3389.3621.3830.4015.4177.4319.4441.4545.4633.4706.4767.4817.4857.4890.4916.4936.4952.4964.4974.4981.4986.4990 * *.00.01.02.03.04.05.06.07.08.09 z Table A-2 Standard Normal ( z ) Distribution

16. 16 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman To find: z Score the distance along horizontal scale of the standard normal distribution; refer to the leftmost column and top row of Table A-2 Area the region under the curve; refer to the values in the body of Table A-2

17. 17 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees.

18. 18 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees. 0 1.58 P ( 0 < x < 1.58 ) =

19. 19 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 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 * *.00.01.02.03.04.05.06.07.08.09 z Table A-2 Standard Normal ( z ) Distribution

20. 20 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees. 0 1.58 Area = 0.4429 P ( 0 < x < 1.58 ) = 0.4429

21. 21 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees. The probability that the chosen thermometer will measure freezing water between 0 and 1.58 degrees is 0.4429. 0 1.58 Area = 0.4429 P ( 0 < x < 1.58 ) = 0.4429

22. 22 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1.58 degrees. There is 44.29% of the thermometers with readings between 0 and 1.58 degrees. 0 1.58 Area = 0.4429 P ( 0 < x < 1.58 ) = 0.4429

23. 23 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Using Symmetry to Find the Area to the Left of the Mean NOTE: Although a z score can be negative, the area under the curve (or the corresponding probability) can never be negative. (a) (b) Because of symmetry, these areas are equal. Equal distance away from 0 0.4925 00 z = 2.43 z = - 2.43 Figure 5-7

24. 24 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water, and if one thermometer is randomly selected, find the probability that it reads freezing water between -2.43 degrees and 0 degrees. The probability that the chosen thermometer will measure freezing water between -2.43 and 0 degrees is 0.4925. -2.43 0 Area = 0.4925 P ( -2.43 < x < 0 ) = 0.4925

25. 25 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman The Empirical Rule Standard Normal Distribution: µ = 0 and  = 1

26. 26 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman x - s x x + sx + s 68% within 1 standard deviation 34% The Empirical Rule Standard Normal Distribution: µ = 0 and  = 1

27. 27 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman x - 2s x - s x x + 2s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations 13.5% The Empirical Rule Standard Normal Distribution: µ = 0 and  = 1

28. 28 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman x - 3s x - 2s x - s x x + 2s x + 3s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean 0.1% 2.4% 13.5% The Empirical Rule Standard Normal Distribution: µ = 0 and  = 1

29. 29 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Probability of Half of a Distribution 0 0.5

30. 30 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Finding the Area to the Right of z = 1.27 0 0.3980 Value found in Table A-2 This area is 0.5 - 0.3980 = 0.1020 z = 1.27 Figure 5-8

31. 31 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Finding the Area Between z = 1.20 and z = 2.30 0 0.3849 0.4893 (from Table A-2 with z = 2.30) Area A is 0.4893 - 0.3849 = 0.1044 z = 1.20 A z = 2.30 Figure 5-9

32. 32 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 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 Notation

33. 33 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Figure 5-10 Interpreting Area Correctly Add to 0.5 x ‘greater than x ’ ‘at least x ’ ‘more than x ’ ‘not less than x ’ x Subtract from 0.5

34. 34 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Figure 5-10 Interpreting Area Correctly Add to 0.5 x Add to 0.5 x ‘greater than x ’ ‘at least x ’ ‘more than x ’ ‘not less than x ’ x Subtract from 0.5 x Subtract from 0.5 ‘less than x ’ ‘at most x ’ ‘no more than x ’ ‘not greater than x ’

35. 35 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Figure 5-10 Interpreting Area Correctly Add to 0.5 x Add to 0.5 x ‘greater than x ’ ‘at least x ’ ‘more than x ’ ‘not less than x ’ x Subtract from 0.5 x Subtract from 0.5 x1x1 x2x2 Add ‘less than x ’ ‘at most x ’ ‘no more than x ’ ‘not greater than x ’ ‘between x 1 and x 2 ’ AB Use A = C - B x1x1 x2x2 C

36. 36 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Finding a z - score when given a probability Using Table A-2 1. Draw a bell-shaped curve, draw the centerline, and identify the region under the curve that corresponds to the given probability. If that region is not bounded by the centerline, work with a known region that is 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.

37. 37 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 0.45 z 0.50 95%5% 5% or 0.05 Finding z Scores when Given Probabilities FIGURE 5-11 Finding the 95th Percentile ( z score will be positive )

38. 38 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 0.45 1.645 0.50 95%5% 5% or 0.05 Finding z Scores when Given Probabilities FIGURE 5-11 Finding the 95th Percentile ( z score will be positive)

39. 39 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 0.40 z 0.10 90% FIGURE 5-12 Finding the 10th Percentile Bottom 10% 10% ( z score will be negative) Finding z Scores when Given Probabilities

40. 40 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 0.40 -1.28 0.10 90% FIGURE 5-12 Finding the 10th Percentile Bottom 10% 10% Finding z Scores when Given Probabilities ( z score will be negative)

41. 41 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Assignment Page 240: 1-36 all