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Chapter Seven Normal Distributions

Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Stolen by: Mistah Flynn. Chapter Seven Normal Distributions. What probability distribution do I use if the variable is not discrete?.

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Chapter Seven Normal Distributions

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  1. Understandable StatisticsSeventh EditionBy Brase and BrasePrepared by: Lynn SmithGloucester County CollegeStolen by: Mistah Flynn Chapter Seven Normal Distributions

  2. What probability distribution do I use if the variable is not discrete? • Discrete variables must be whole, counting numbers like numbers on a number line • Continuous variables can be whole, fractional, or decimal numbers like number in an interval

  3. So what is the difference? Discrete Continuous

  4. The Normal Distribution

  5. Properties of The Normal Distribution  The curve is bell-shaped with the highest point over the mean, .

  6. Properties of The Normal Distribution  The curve is symmetrical about a vertical line through .

  7. Properties of The Normal Distribution  –  The transition points between cupping upward and downward occur above  +  and  –  .

  8. The Normal Density Function This formula generates the density curve which gives the shape of the normal distribution.

  9. The Empirical Rule Approximately 68% of the data values lie is within one standard deviation of the mean. 68% One standard deviation from the mean.

  10. The Empirical Rule Approximately 95% of the data values lie within two standard deviations of the mean. 95% Two standard deviations from the mean.

  11. The Empirical Rule Almost all (approximately 99.7%) of the data values will be within three standard deviations of the mean. 99.7% Three standard deviations from the mean.

  12. The beauty of the normal curve?: The empirical rule! No matter what  and  are, the area between - and + is about 68%; the area between -2 and +2 is about 95%; and the area between -3 and +3 is about 99.7%. Almost all values fall within 3 standard deviations.

  13. 68% of the data 95% of the data 99.7% of the data 68-95-99.7 Rule

  14. 68-95-99.7 Rulein 5th grade terms… • Draw a runway or skateboard alley (gotta roll…) • Then draw a plane or skateboard taking off and landing (flight path…not really quadratic is it…) • Split the pathway in half with a vertical line (folding does work ; ) • Don’t’ bottom out; leave a little room in each tail and draw two more vertical lines • Split the distance between the ends and the middle (Split the uprights! Between the right and left vertical lines) • Label in the graph your Empirical Rule values (68-95-99.7) • Label below the graph a triple x axis = Z score, %ile, and Empirical splits or x values • Add any other statistics in context…

  15. 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%

  16. Control Chart a statistical tool to track data over a period of equally spaced time intervals or in some sequential order

  17. Statistical Control A random variable is in statistical control if it can be described by the same probability distribution when it is observed at successive points in time.

  18. To Construct a Control Chart • Draw a center horizontal line at . • Draw dashed lines (control limits) at +, -, +, and -. • The values of  and  may be target values or may be computed from past data when the process was in control. • Plot the variable being measured using time on the horizontal axis.

  19. Control Chart      1 2 3 4 5 6 7

  20. Control Chart      Day – value 1 25 2 26.1 3 24 4 16.4 5 25.5 6 14.3 7 41 1 2 3 4 5 6 7

  21. Out-Of-Control Warning Signals I One point beyond the 3 level II A run of nine consecutive points on one side of the center line at target  III At least two of three consecutive points beyond the 2 level on the same side of the center line.

  22. Probability of a False Alarm

  23. 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.

  24. Formula for z score

  25. 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.

  26. Calculating z-scores Mean delivery time = 25 minutes Standard deviation = 2 minutes Convert 29.7 minutes to a z score.

  27. 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.

  28. Standard Normal Distribution:  = 0  = 1 -1 1 0 Values are converted to z scores where z =

  29. Importance of the Standard Normal Distribution: Standard Normal Distribution: 1 Any Normal Distribution: 0 Areas will be equal. 1 

  30. Use of the Normal Probability Table (Table 4) - 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.

  31. Use of the Normal Probability Table (Table 4a) - 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.

  32. To find the area to the left of z = 1.34 _____________________________________z … 0.03 0.04 0.05 ..… _____________________________________ . . 1.2 … .8907 .8925 .8944 …. 1.3 … .9082 .9099 .9115 …. 1.4 … .9236 .9251 .9265 …. .

  33. Patterns for Finding Areas Under the Standard Normal Curve To find the area to the left of a given negative z (less than or equal to): Use Table 4 (Appendix II) directly. z 0

  34. Patterns for Finding Areas Under the Standard Normal Curve To find the area to the left of a given positive z (less than or equal to): Use Table 5 a (Appendix II) directly. z 0

  35. Patterns for Finding Areas Under the Standard Normal Curve To find the area between z values on either side of zero (interval between values): Subtract area to left of z1 from area to left of z2 . z2 0 z1

  36. Patterns for Finding Areas Under the Standard Normal Curve To find the area between z values on the same side of zero (interval of values): Subtract area to left of z1 from area to left of z2 . z1 z2 0

  37. 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 (greater than or equal to) : Subtract from 1.0000 the area to the left of the given z. Area under entire curve = 1.000. z 0

  38. 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

  39. Normal Probability .9974 d. P( - 3 < z < 3 ) = ________ e. P( - 2.34 < z < 1.57 ) = _____ f. P( 1.24 < z < 1.88 ) = _______ .9322 .0774

  40. Normal Probability .2254 g. P( - 2.44 < z < - 0.73 ) = _______ h. P( z < 1.64 ) = __________ i . P( z > 2.39 ) = _________ .9495 .0084

  41. Normal Probability j. P ( z > - 1.43 ) = __________ k. P( z < - 2.71 ) = __________ .9236 .0034

  42. 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

  43. Inverse Normal Distribution Finding z scores when probabilities (areas) are given

  44. Find the indicated z score: Find the indicated z score: .8907 0 z = 1.23

  45. z 0 Find the indicated z score: .6331 .3669 z = – 0.34

  46. Find the indicated z score: .3560 .8560 0 z = 1.06

  47. Find the indicated z score: .4792 .0208 – 2.04 z = 0

  48. Find the indicated z score: .4900 0 z = 2.33

  49. Find the indicated z score: .005 z = 0 – 2.575

  50. Find the indicated z score: = .005 A B – z 0 z  2.575 or  2.58 If area A + area B = .01, z = __________

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