2. Unit 4 Randomness in Data Topic 15 Normal Distributions (page 325)

3. OVERVIEW You began studying randomness and probability in the previous topic. Toward the end of that topic you saw that a mound-shaped distribution described the outcomes of a particular variable, such as counts and proportions. Such a pattern arises often enough that it has been very extensively studied mathematically.

4. OVERVIEW In this topic you will investigate mathematical models known as normal distributions, which describe this pattern of variation very accurately. You will learn how to use normal distributions to calculate probabilities of interest in a variety of contexts.

5. Do the Preliminaries (page 326)

6. Essential Question How can normal curves be use as mathematical models for approximating distributions?

7. Activity 15-1 Placement Scores and Hospital Births (page 327)

8. (a)The similarities that I notice about the shapes of these distributions is that they are … (b)[Draw a sketch of a smooth curve that approximates the general shape of the histograms.] roughly symmetric, mound shaped, with a single peak.

10. Data that display the general shape shown above occur frequently. Theoretical mathematical models used to approximate such distributions are called normal distributions.

11. Each normal distribution shares 3 characteristics: (1) Every normal distribution is symmetric. (2) Every normal distribution has a single peak at its center. (3) Every normal distribution follows a bell-shaped curve.

12. Each normal distribution is distinguished (identified) by 2 things: (1) its mean and (2) its standard deviation.

13. The mean µ determines where its center is; the peak of a normal curve is at its mean and its point of symmetry. The standard deviation indicates how spread out the distribution is. The distance between the mean µ and the points where the curvature changes is equal to the standard deviation.

14. Greek Alphabet

18. (c) [Label each curve with A, B, or C.] A B C

19. (d)Are these proportions quite close to each other and to the predictions of the empirical rule that you studied in Topic 5? _______ YES

20. Review: (e)z = _____________________ = _________ (show work with answer rounded to the nearest hundredth) (f) z = _____________________ = _________ (show work with answer rounded to the nearest hundredth) -0.45

21. (g)The thing true about these 2 scores is that they are equal. (h)The interpretation of these z-scores is that the two values fall 0.45 of one standard deviation below their respective means.

22. (i)The proportion of observations falling below 8.5 in the placement exam data is _________. 33%

23. (j)The proportion of observations falling below 23.5 in the simulated births data is _________. 32%

24. The proportion of observations falling below 8.5 in the placement exam data is 33%. The proportion of observations falling below 23.5 in the simulated births data is 32%. Are these proportions fairly close? ______ YES

25. The closeness of these percentages indicates that to find the proportion of data falling in a given region for normal distributions, all we need to determine is the z-score. Values with the same z-score will have the same percentage of observations lying below them, for any normal distribution. Thus, instead of finding percentages for all normal distributions, we need only the percentages corresponding to these z-scores.

26. (k)The probability that the randomly selected student’s score will be below 8.5 is _________. [Hint: Refer back to your answer to (i).] 33%

27. The probability of a randomly selected observation falling in a certain interval is equivalent to the proportion of the population’s observations falling in that interval. Since the total area under the curve of a normal distribution is 1, this probability can be calculated by finding the area under the normal curve for that interval. The total area under the curve is always equal to 1 or 100%.

30. (l)Area corresponding to the probability of a placement score falling below 8.5 = … Pr(z<-.45)

32. (l)Area corresponding to the probability of a placement score falling below 8.5 = … (m)Is this value reasonably close to your answers for the proportions of the placement and birth data less than their z-scores of -0.45?_______ =.3264Pr(z<-.45) YES = 33%

33. Use Z to denote the standard normal distribution. The notation Pr ( a < Z < b ) denotes the probability lying between the values a and b, calculated as the area under the standard normal curve in that region. The notation Pr ( Z ≤ z ) denotes the area to the left of a particular value z, while Pr( Z ≥ z ) refers to the area above a particular z value.

34. EXAMPLES Sketch the curve and the area under the curve, then find the proportion. Use the Standard Normal Probability Table from your notes.

35. EXAMPLE #1 Sketch the curve and the area under the curve, then find the proportion. Use the Standard Normal Probability Table from your notes. Pr (Z < 0.68) = _________ = _________.751775.17% 0.68 Less than sign means shade to the left.

36. EXAMPLE #2 Sketch the curve and the area under the curve, then find the proportion. Use the Standard Normal Probability Table from your notes. Pr (Z ≤ 0.68) = _________ = _________.751775.17% 0.68 The less than sign and less than or equal to sign are treated the same.

37. EXAMPLE #3 Sketch the curve and the area under the curve, then find the proportion. Use the Standard Normal Probability Table from your notes. Pr (Z > 0.68) = _________ = _________.248324.83% 0.68 Greater than sign means shade to the right. 100% - 75.17% = 24.83%

38. EXAMPLE #3 Here is another way to look at this problem! Pr (Z > 0.68) = _________ = _________.248324.83% - 0.68 Remember … Normal curves are symmetric! You can use the table if you think of the opposite Pr (Z < -0.68).

39. EXAMPLE #4 Sketch the curve and the area under the curve, then find the proportion. Use the Standard Normal Probability Table from your notes. Pr (Z < -1.38) = _________ = _________.08388.38% -1.38

40. EXAMPLE #5 Sketch the curve and the area under the curve, then find the proportion. Use the Standard Normal Probability Table from your notes. Pr (Z > -1.75) = _________ = _________.959995.99% -1.75

41. EXAMPLE #6 Sketch the curve and the area under the curve, then find the proportion. Use the Standard Normal Probability Table from your notes. Pr (Z < -2.49) = _________ = _________.00640.64% -2.49

42. EXAMPLE #7 Sketch the curve and the area under the curve, then find the proportion. Use the Standard Normal Probability Table from your notes. Pr (-1.38 < Z < 0.68) = _______ = ________ -1.38 0.68

43. Find the area of the smaller region … Pr (Z < -1.38) = _________ = _________.08388.38% -1.38

44. Then find the area of the larger region … Pr (Z < 0.68) = _________ = _________.751775.17% 0.68

45. Subtract the area of the smaller region from the area of the larger region … Pr (-1.38 < Z < 0.68) = Pr (Z < 0.68) - Pr (Z < -1.38) =.7517 -.0838 -1.38

46. EXAMPLE #7 Pr (-1.38 < Z < 0.68) = ________ = __________.667966.79% -1.38 0.68

47. EXAMPLE #8 Sketch the curve and the area under the curve, then find the proportion. Use the Standard Normal Probability Table from your notes. Pr (Z < -3.81) = ____________________ less than.0002 -3.81

48. EXAMPLE #9 Sketch the curve and the area under the curve, then find the z-value. Use the Standard Normal Probability Table from your notes in reverse. Pr (Z < k) =.8997  k = ________ 1.28 ? Must find the Z-value. 89.97%

49. EXAMPLE #10 Sketch the curve and the area under the curve, then find the z-value. Use the Standard Normal Probability Table from your notes in reverse. Pr (Z > k) =.0630  k = ________ ? 1.53 100% - 6.30% = 93.70% So lookup.9370.

50. Take-Home quiz on these examples.

51. Essential Question How can normal curves be use as mathematical models for approximating distributions?

52. mean µ = 3250 grams and standard deviation  = 550 grams (a) Note: Low birth weight is under 2500 grams. Activity 15-2 Birth Weights (pages 331 to 334)

53. Remember: Low birth weight is under 2500 grams. low birth weight is under 2500 grams

54. b)My guess as to the proportion of babies born with a low birth weight is _________.

55. (c) z-score = ________________ = ______ (show work with answer rounded to the nearest hundredth) (d)Pr ( Z < __________) = ___________ = ____________ (e) [Follow directions with the calculator.] Does your answer match (to within rounding discrepancies) your answer to (d)?_________ -1.36 YES -1.36.08698.69% normalcdf (-1E99, 2500, 3250, 550) = lower bound upper bound mean μ s.d. σ. 0863

56. (f)[Use Standard Normal Probabilities Table.] z-score = ________________ = ______ (show work with answer rounded to the nearest hundredth) Pr ( Z > __________) = ___________ = ____________ 2.34.00960.96%

57. [Use your calculator.] Pr ( X > 4536) = ___________ = ____________.00970.97% normalcdf (4536, 1E99, 3250, 550) lower bound upper bound mean μ s.d. σ

58. (g) [Know both ways in (f).] Pr ( Z > 2.34) = 0.96% Pr ( X > 4536) = 0.97%

59. (h) mean µ = 3250 grams & standard deviation  = 550 grams lower bound 3000 upper bound 4000

60. z-score = ________________ = ______ Pr (__________ < Z < __________) = ___________ - ____________ = _____________ [Use Standard Normal Probabilities Table.] 1.36 -0.45 1.36.9131.3264 58.67%

61. [Use your calculator.] Pr ( 3000 < X < 4000 ) = ___________ = _____________ normalcdf (3000, 4000, 3250, 550) lower bound upper bound mean μ s.d. σ.588958.89%

62. (i)Proportion for low birth weight babies is __________. Proportion for babies weighing between 3000 and 4000 is __________. 7.50% 65.78%

63. [comment] First of all I just want to point out that my youngest son, Noah, was born in 1997. The proportion for low birth weight babies was off by about 1% (8.69% vs. 7.50%). The proportion of babies weighing between 3000 and 4000 grams is off by less than 7% (58.89% vs 65.78%).

64. (j)A newborn would have to weigh _______ grams to be among the lightest 2.5%. invNorm (.025, 3250, 550) % to left of value mean μ s.d. σ 2172 2.5% 2172 grams ?

65. (k)A newborn would have to weigh _______ grams to be among the heaviest 10%. invNorm (.9, 3250, 550) % to left mean μ s.d. σ 3955 10% 90% ? 3955 grams

66. Assignment Activity 15-5: Pregnancy Durations (page 338) Assignment Activity 15-6: Professors’ Grades (pages 338 & 339) Assignment Activity 15-8: IQ Scores (page 339) How can the properties of a normal distirbution be used to make predictions?

67. Essential Question How can normal curves be use as mathematical models for approximating distributions?

68. Activity 15-3 Matching Samples to Density Curves (pages 334 to 336) While normal distributions are the most common, they are not the only kind of theoretical probability model. Any curve under which the total area is one and for which areas correspond to probabilities represents a probability model. Such curves are called density curves.

70. (a)Sample 1: Population _______ Sample 2: Population _______ Sample 3: Population _______ Sample 4: Population _______ A B C D

71. 1 2 3 4 (a)Here are the answers again.

72. (b)Now try it with a sample size of 10!

73. (b)Sample 5: Population _______ Sample 6: Population _______ Sample 7: Population _______ Sample 8: Population _______ C B D A

74. (c) It is easier to discern the shape of the population from a sample size of ______. 100

75. Especially with small sample sizes, sample data from normal populations may not look very normal and may be hard to distinguish from sample data from other shapes of populations.

76. WRAP-UP This topic has introduced you to the most important mathematical model in all of statistics: the normal distribution. You have discovered how z-scores provide the key to using a table of standard normal probabilities to perform calculations related to normal distributions. You have also compared predictions from a normal model to observed data as a means of assessing the usefulness of the model in a given situation. You have seen that the model predictions are not exact, but are close, especially with larger samples.

77. The next two topics will reveal how the normal distributions describe the pattern of variation that arises when one repeatedly takes samples from a population. In the next topic you will explore how sample proportions vary, while the variation of sample means will occupy your attention in Topic 17. These topics point toward the key role of normal distributions in the most important theoretical result in all of statistics: the Central Limit Theorem.

78. Activity 15-4 Normal Curves (pages 337 to 338) a: mean µ = _______ and standard deviation  = _______ b: mean µ = _______ and standard deviation  = _______ c: mean µ = _______ and standard deviation  = _______ d: mean µ = _______ and standard deviation  = _______ Click for HELP!

79. Remember : The mean is in the middle of the curve … and the normal curve is contained within 3 standard deviations from the mean! a: mean µ = _______ and standard deviation  = _______ 35 45 5565 505

80. Remember : The mean is in the middle of the curve … and the normal curve is contained within 3 standard deviations from the mean! b: mean µ = ________ and standard deviation  = _______ 0 1000 2000 1100300

81. c: mean µ = _______ and standard deviation  = _______ -100 0100 -1545 Remember : The mean is in the middle of the curve … and the normal curve is contained within 3 standard deviations from the mean!

82. Now you do this one on your own. d: mean µ = _______ and standard deviation  = _______ 0100300500 22575 200400

83. Assignment Activity 15-14: Empirical Rule (page 341) Click for HELP!

84. 68% 95% 99.7% Click for another assignment!

85. Assignment Activity 15-15: Critical Values (page 342) Here is some help to get you started!

86. Activity 15-15: Critical Values (page 342) (a) Pr(Z>z*) =.10  then z* = _______ invNorm (.9, 0, 1 ) or just invNorm (.9) % to left mean μ s.d. σ 10% 90% z*z* 1.28 Since it is standardized!

87. 1.28 1.645 1.96 2.33 2.58 These values are called the critical values of the normal distribution.

88. Your topic is due! The quiz for this topic is on the facts of a Normal Distribution. How can the properties of a normal distirbution be used to make predictions?