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  z Score (or standard score) the number of standard deviations that a given value x is above or below the mean Measures of Position (Section 2.6)

3. 3 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Sample z = x - x s Population z = x - µ  Round to 2 decimal places Measures of Position z score

4. 4 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman - 3- 2- 101 23 Z Unusual Values Unusual Values Ordinary Values Interpreting Z Scores FIGURE 2-16

5. 5 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Chapter 5 Normal Probability Distributions 5-1 Overview 5-2 The Standard Normal Distribution 5-3 Normal Distributions: Finding Probabilities 5-4 Normal Distributions: Finding Values 5-5The Central Limit Theorem 5-6 Normal Distribution as Approximation to Binomial Distribution (skipped) 5-7Determining Normality

6. 6 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman  Continuous random variable  Graph is symmetric and bell shaped  Area under the curve is equal to 1, therefore there is correspondence between area and probability. µ Normal Distribution

7. 7 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 Normal distributions (bell shaped) are a family of distributions that have the same general shape. The mean, mu, controls the center Standard deviation, sigma, controls the spread.

8. 8 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5-2 The Standard Normal Distribution

9. 9 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition Standard Normal Distribution a normal probability distribution that has a mean of 0 and a standard deviation of 1

10. 10 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman -3-2 0 231 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 z

11. 11 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Example: Using your knowledge of the Emperical rule, what is the probability that a value falls between the mean and 1 standard deviation above the mean?

12. 12 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Example: Using your knowledge of the Emperical rule, what is the probability that a value falls between the mean and 1 standard deviation above the mean? 1 34% 0z

13. 13 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman What if we need to find the probability of a value that does not fall on a standard deviation?

14. 14 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Step 1: Draw a normal curve with the centerline labeled 0 on the x-axis. Step 2: Label given value(s) on the appropriate location on the x-axis. Draw vertical lines on the normal curve above the given values. Step 3: Shade the area of the region in question. Step 4: Use calculator function normalcdf() to find the area (probability) in question. normalcdf(left boundary, right boundary) Finding Probabilities from Standard(Z) scores

15. 15 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. Since  = 0 and  =1, the values are standard or z-scores.

16. 16 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 1.58 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. Since  = 0 and  =1, the values are standard or z-scores.

17. 17 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 1.58 P ( 0 < x < 1.58 ) = ? 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. Since  = 0 and  =1, the values are standard or z-scores.

18. 18 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

19. 19 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 1.58 P ( 0 < x < 1.58 ) = Normalcdf(0, 1.58) =.443 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. Since  = 0 and  =1, the values are standard or z-scores.

20. 20 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman The probability that the chosen thermometer will measure freezing water between 0 and 1.58 degrees is 0.443. 0 1.58 Area = 0.443 P ( 0 < x < 1.58 ) = 0.443 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. Since  = 0 and  =1, the values are standard or z-scores.

21. 21 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman There is 44.3% of the thermometers with readings between 0 and 1.58 degrees. 0 1.58 Area = 0.443 P ( 0 < x < 1.58 ) = 0.443 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. Since  = 0 and  =1, the values are standard or z-scores.

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 -2.43 degrees and 0 degrees. -2.43 0

23. 23 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. -2.43 0 Area = 0.493 P ( -2.43 < x < 0 ) = Normalcdf(-2.43, 0)= 0.493 NOTE: Although a z score can be negative, the area under the curve (or the corresponding probability) can never be negative.

24. 24 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman P(a < z < b) → Normalcdf(a,b) denotes the probability that the z score is between a and b P( z > a) → Normalcdf(a,9999) denotes the probability that the z score is greater than a P ( z < a) → Normalcdf(-9999,a) denotes the probability that the z score is less than a Note: 9999 is used to represent infinity Notation

25. 25 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 0.5 Remember the graph represents the entire distribution with its area of 1.

26. 26 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 P(z>1.27) = normalcdf(1.27, 9999) = 0.1020 z = 1.27

27. 27 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 z = 1.20 Area =.104 z = 2.30 P(1.20 <z< 2.30) = normalcdf(1.20, 2.30) = 0.104

28. 28 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Figure 5-10 Interpreting Area Correctly x ‘greater than x ’ ‘at least x ’ ‘more than x ’ ‘not less than x ’ x

29. 29 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman ‘less than x ’ ‘at most x ’ ‘no more than x ’ ‘not greater than x ’ Interpreting Area Correctly x x ‘greater than x ’ ‘at least x ’ ‘more than x ’ ‘not less than x ’ x x

30. 30 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 1. Draw a bell-shaped curve, draw the centerline, and shade the region under the curve that corresponds to the given probability. The line(s) that bound the region denotes the z- score(s) you are trying to find. 2.Using the probability representing the area bounded by the z-score, convert it to area to the left of the z-score and enter that value into invNorm() function in your calculator. z = invNorm(area to the left) This command finds the z-score that has area to the left of the boundary under the normal curve. Since the area represent probability the argument must be between 0 and 1 inclusive. 3.If the z score is positioned to the left of the centerline, make sure it’s negative.

31. 31 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman ( z score will be positive ) 0 0.95 z 95% Finding z Scores when Given Probabilities Find the z-score that denotes the 95th Percentile. z = invNorm(.95) =1.65

32. 32 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 0.90 z Find the z-score that separate the bottom 10% and upper 90%. 0.10 90% Bottom 10% 10% ( z score will be negative) Finding z Scores when Given Probabilities z = invNorm(.10) = -1.28

33. 33 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 0 60% 0.60 z = invNorm(.20) = -0.84 Find the z-scores that denotes the middle 60% Finding z Scores when Given Probabilities 0.20 Bottom 20% 20% 0.20 20% z = invNorm(.80) = 0.84