1. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 1 The Normal Distribution

2. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 2 Properties of The Normal Distribution The curve is bell-shaped with the highest point over the mean, . 

3. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 3 Properties of The Normal Distribution The curve is symmetrical about a vertical line through . 

4. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 4 Properties of The Normal Distribution The curve approaches the horizontal axis but never touches or crosses it. 

5. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 5 Properties of The Normal Distribution The transition points between cupping upward and downward occur above  +  and  – .  –     

6. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 6 The Empirical Rule Approximately 68% of the data values lie is within one standard deviation of the mean. One standard deviation from the mean. 68%

7. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 7 The Empirical Rule Approximately 95% of the data values lie within two standard deviations of the mean. Two standard deviations from the mean. 95%

8. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 8 The Empirical Rule Almost all (approximately 99.7%) of the data values will be within three standard deviations of the mean. Three standard deviations from the mean. 99.7%

9. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 9 Application of the Empirical Rule The life of a particular type of light bulb is normally distributed with a mean of 1100 hours and a standard deviation of 100 hours. What is the probability that a light bulb of this type will last between 1000 and 1200 hours? Approximately 68%

10. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 10 Z Score The z value or z score tells the number of standard deviations the original measurement is from the mean. The z value is in standard units.

11. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 11 Formula for z score

12. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 12 Calculating z-scores The amount of time it takes for a pizza delivery is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. Convert 21 minutes to a z score.

13. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 13 Calculating z-scores Mean delivery time = 25 minutes Standard deviation = 2 minutes Convert 29.7 minutes to a z score.

14. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 14 Interpreting z-scores Mean delivery time = 25 minutes Standard deviation = 2 minutes Interpret a z score of 1.6. The delivery time is 28.2 minutes.

15. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 15 Standard Normal Distribution:  = 0  = 1 1 Values are converted to z scores where z = 0

16. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 16 Importance of the Standard Normal Distribution: 1 0 11  Areas will be equal. Any Normal Distribution: Standard Normal Distribution:

17. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 17 Use of the Normal Probability Table (Table 5) - Appendix II Entries give the probability that a standard normally distributed random variable will assume a value to the left of a given negative z-score.

18. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 18 Use of the Normal Probability Table (Table 5a) - Appendix II Entries give the probability that a standard normally distributed random variable will assume a value to the left of a given positive z value.

19. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 19 To find the area to the left of z = 1.34 _____________________________________ z…0.030.040.05..… _____________________________________. 1.2….8907.8925.8944 …. 1.3 ….9082.9099.9115 …. 1.4….9236.9251.9265 …..

20. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 20 Patterns for Finding Areas Under the Standard Normal Curve To find the area to the left of a given negative z : Use Table 5 (Appendix II) directly. z 0

21. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 21 Patterns for Finding Areas Under the Standard Normal Curve To find the area to the left of a given positive z : Use Table 5 a (Appendix II) directly. z 0

22. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 22 Patterns for Finding Areas Under the Standard Normal Curve To find the area between z values on either side of zero: Subtract area to left of z 1 from area to left of z 2. z2z2 0 z1z1

23. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 23 Patterns for Finding Areas Under the Standard Normal Curve To find the area between z values on the same side of zero: Subtract area to left of z 1 from area to left of z 2. z2z2 0 z1z1

24. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 24 Patterns for Finding Areas Under the Standard Normal Curve To find the area to the right of a positive z value or to the right of a negative z value: Subtract from 1.0000 the area to the left of the given z. z 0 Area under entire curve = 1.000.

25. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 25 Use of the Normal Probability Table a.P(z < 1.24) = ______ b. P(0 < z < 1.60) = _______ c.P( - 2.37 < z < 0) = ______.8925.4452.4911

26. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 26 Normal Probability d.P( - 3 < z < 3 ) = ________ e. P( - 2.34 < z < 1.57 ) = _____ f.P( 1.24 < z < 1.88 ) = _______.9974.9322.0774

27. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 27 Normal Probability g. P( - 2.44 < z < - 0.73 ) = _______ h.P( z < 1.64 ) = __________ i. P( z > 2.39 ) = _________.9495.0084.2254

28. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 28 Normal Probability j.P ( z > - 1.43 ) = __________ k. P( z < - 2.71 ) = __________.9236.0034

29. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 29 Application of the Normal Curve The amount of time it takes for a pizza delivery is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. If you order a pizza, find the probability that the delivery time will be: a.between 25 and 27 minutes.a. ___________ b.less than 30 minutes.b. __________ c.less than 22.7 minutes.c. __________.3413.9938.1251

30. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 30 Inverse Normal Distribution Finding z scores when probabilities (areas) are given

31. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 31 Find the indicated z score:.8907 0 z = 1.23

32. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 32 Find the indicated z score:.6331 z 0 z = – 0.34.3669

33. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 33 Find the indicated z score:.3560 0 z = 1.06.8560

34. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 34 Find the indicated z score:.4792 z = 0 – 2.04.0208

35. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 35 Find the indicated z score: 0 z =.4900 2.33

36. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 36 Find the indicated z score: z = 0.005 – 2.575

37. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved. 37 Find the indicated z score: If area A + area B =.01, z = __________ A B – z 0 z  2.575 or  2.58 =.005