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Understanding Normal Distribution Properties and Applications

Learn about the properties and applications of the normal distribution, including the Empirical Rule, standard scores, raw scores, and the importance of the standard normal distribution. Explore how to work with any normal distribution and apply the Normal Curve to solve real-world problems accurately.

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Understanding Normal Distribution Properties and Applications

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  1. Chapter Seven Normal Curves and Sampling Distributions

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

  3. Properties of The Normal Distribution The curve is symmetrical about a vertical line through μ.

  4. Properties of The Normal Distribution The curve approaches the horizontal axis but never touches or crosses it.

  5. Properties of The Normal Distribution The transition points between cupping upward and downward occur above μ + σ and μ – σ .

  6. The Empirical Rule • Applies to any symmetrical and bell-shaped distribution

  7. The Empirical Rule Approximately 68% of the data values lie within one standard deviation of the mean.

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

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

  10. 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%

  11. Standard Score • The z value or z score tells the number of standard deviations between the original measurement and the mean. • The z value is in standard units.

  12. Formula for z score

  13. x Values and Corresponding z Values

  14. 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.

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

  16. Raw Score • A raw score is the result of converting from standard units (z scores) back to original measurements, x values. • Formula: x = z s + m

  17. Interpreting z-scores Mean delivery time = 25 minutes Standard deviation = 2 minutes Interpret a z score of 1.60. The delivery time is 28.2 minutes.

  18. Standard Normal Distribution: m= 0 s= 1 Any x values are converted to z scores.

  19. Importance of the Standard Normal Distribution: Any Normal Curve: Areas will be equal.

  20. Areas of a Standard Normal Distribution • Appendix • Table 3 • Pages A6 - A7

  21. Use of the Normal Probability Table Appendix Table 3 is a left-tail style table. Entries give the cumulative areas to the left of a specified z.

  22. To Find the area to the Left of a Given z score • Find the row associated with the sign, units and tenths portion of z in the left column of Table 3. • Move across the selected row to the column headed by the hundredths digit of the given z.

  23. Find the area to the left of z = – 2.84

  24. To find the area to the left of z = – 2.84

  25. To find the area to the left of z = – 2.84

  26. The area to the left of z = – 2.84 is .0023

  27. To Find the Area to the Left of a Given Negative z Value: Use Table 3 of the Appendix directly.

  28. To Find the Area to the Leftof a Given Positive z Value: Use Table 3 of the Appendix directly.

  29. To Find the Area to the Rightof a Given z Value: Subtract the area to the left of z from 1.0000.

  30. Alternate Way To Find the Area to the Right of a Given Positive z Value: Use the symmetry of the normal distribution. Area to the right of z = area to left of –z.

  31. To Find the Area Between Two z Values Subtract area to left of z1 from area to left of z2 . (When z2 > z1.)

  32. Convention for Using Table 3 • Treat any area to the left of a z value smaller than 3.49 as 0.000 • Treat any area to the left of a z value greater than 3.49 as 1.000

  33. Use of the Normal Probability Table a. P( z < 1.64 ) = __________ b. P( z < - 2.71 ) = __________ .9495 .0034

  34. Use of the Normal Probability Table c. P(0 < z < 1.24) = ______ d. P(0 < z < 1.60) = _______ e. P( 2.37 < z < 0) = ______ .3925 .4452 .4911

  35. Use of the Normal Probability Table .9974 f. P( 3 < z < 3 ) = ________ g. P( 2.34 < z < 1.57 ) = _____ h. P( 1.24 < z < 1.88 ) = _______ .9322 .0774

  36. Use of the Normal Probability Table i. P( 2.44 < z < 0.73 ) = _______ j . P( z > 2.39 ) = ________ k. P( z > 1.43 ) = __________ .2254 .0084 .9236

  37. To Work with Any Normal Distributions • Convert x values to z values using the formula: Use Table 3 of the Appendix to find corresponding areas and probabilities.

  38. Rounding • Round z values to the hundredths positions before using Table 3. • Leave area results with four digits to the right of the decimal point.

  39. 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: • between 25 and 27 minutes. a. __________ • less than 30 minutes. b. __________ • c. less than 22.7 minutes. c. __________ .3413 .9938 .1251

  40. Inverse Normal Probability Distribution • Finding z or x values that correspond to a given area under the normal curve

  41. Inverse Normal Left Tail Case Look up area A in body of Table 3 and use corresponding z value.

  42. Inverse Normal Right Tail Case: Look up the number 1 – A in body of Table 3 and use corresponding z value.

  43. Inverse Normal Center Case: Look up the number (1 – A)/2 in body of Table 3 and use corresponding ± z value.

  44. Using Table 3 for Inverse Normal Distribution • Use the nearest area value rather than interpolating. • When the area is exactly halfway between two area values, use the z value exactly halfway between the z values of the corresponding table areas.

  45. When the area is exactly halfway between two area values • When the z value corresponding to an area is smaller than 2, use the z value corresponding to the smaller area. • When the z value corresponding to an area is larger than 2, use the z value corresponding to the larger area.

  46. Find the indicated z score:

  47. Find the indicated z score: – 2.57 z = _______

  48. Find the indicated z score: 2.33 z = _______

  49. Find the indicated z scores: –1.23 1.23 –z = _____ z = ____

  50. Find the indicated z scores: ± 2.58 ± z =__________

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