1. Chapter 6 The Normal Distribution

2.  The Normal Distribution  The Standard Normal Distribution  Applications of Normal Distributions  Sampling Distributions  The Central Limit Theorem (CLT)

3. Probability Distributions of Continuous random variables

4. Recall

5. Probability Distribution of Total Number of Right Answers

6. Recall

7. Probability Distribution of Total Number of Right Answers

8. A continuous random variable has an infinite number of possible values that can be represented by an interval on the number line. For example Average Grade in Math 1111 04020609080100 Average Grade can be any number between 0 and 100. The probability distribution of a continuous random variable is called a continuous probability distribution. Continuous Probability Distribution

9. Relative Frequency Histogram Average Final Grade for 60 students

10. Average Final Grade for 450 students Relative Frequency Histogram

11. Average Final Grade for 1260 students Relative Frequency Histogram

12. Average Final Grade for 1961 students Relative Frequency Histogram

13. Average Final Grade for 1961 students Relative Frequency Histogram

14. Average Final Grade for all students Continuous Probability Distribution

15. Data World Theory World

16. It is mathematically easier to work with such a curve instead of a histogram Density Curve or Density Function Mathematical Model

17. A density curve is a curve that 1) is always on or above the horizontal axis. 2) has area exactly equal to 1 underneath it. Density Curve or Density Function A density curve describes the overall pattern of a continuous distribution. The area under the curve and above any range of values is the relative frequency of all observations that fall in that range (see picture in next slide).

18. X Shaded area equals percentage of observations that fall between a and b P(a  X  b) is equal to this area ba Density Curve or Density Function

19. Examples of Density Functions

20. Normal Distributions and the Standard Distribution

21. Properties of Normal Distributions The most important probability distribution in statistics is the normal distribution. A normal distribution is a continuous probability distribution for a random variable, X. The graph of a normal distribution is called the normal curve. Normal curve x

22. 1) The mean, median, and mode are equal. 2) The normal curve is bell-shaped and symmetric about the mean. 3) The total area under the curve is equal to one. 4) The normal curve approaches, but never touches the x-axis as it extends farther and farther away from the mean. 5) Between μ  σ and μ + σ the graph curves downward. The graph curves upward to the left of μ  σ and to the right of μ + σ. Properties of Normal Distributions

23. μ  3σ μ + σ μ  2σμ  σμ  σ μμ + 2σμ + 3σ Inflection points Total area = 1 If X is a continuous random variable having a normal distribution with mean μ and standard deviation σ, you can graph a normal curve with the equation x Properties of Normal Distributions

24. The following table displays a frequency and a relative- frequency distribution for the heights of the 3264 female students who attend a Midwestern college. Example of Normal Distribution

25. The figure displays a relative-frequency histogram for the same data. Included is a shaded area between 67” and 68”. This area is P(67  Height  68) = 0.0735

26. We now included a normal curve that approximates the overall shape of the distribution. The area under the curve. between 67 and 68 is P(67  Height  68) = 0.0735

27. Means and Standard Deviations A normal distribution can have any mean and any positive standard deviation. Mean: μ = 3.5 Standard deviation: σ  1.3 Mean: μ = 6 Standard deviation: σ  1.9 The mean gives the location of the line of symmetry. The standard deviation describes the spread of the data. Inflection points 361542 x 3615429711108 x

28. Means and Standard Deviations Example: 1.Which curve has the greater mean? 2.Which curve has the greater standard deviation? The line of symmetry of curve A occurs at x = 5. The line of symmetry of curve B occurs at x = 9. Curve B has the greater mean. Curve B is more spread out than curve A, so curve B has the greater standard deviation. 315971113 A B x

29. 33 1 22 11 02 3 z The Standard Normal Distribution The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The horizontal scale corresponds to z - scores. If a variable X has a normal distribution with mean μ and standard deviation σ, then the standardized variable has the standard normal distribution

30. z = 3.49 Area is close to 1. z = 0 Area is 0.5000. Properties of the Standard Normal Distribution 1. The cumulative area is close to 0 for z - scores close to z =  3.49. 2. The cumulative area increases as the z - scores increase. 3. The cumulative area for z = 0 is 0.5000. 4. The cumulative area is close to 1 for z - scores close to z = 3.49 z =  3.49 Area is close to 0. z 33 1 22 11 023

31. Areas Under the Standard Normal Curve

32. P(Z  z) = The Standard Normal Table

34. Example - Find P(Z  1.40) P(Z  1.40)=0.9192

35. Finding P(Z  1.23), that is, finding the area under the standard normal curve to the left of 1.23 Always draw the curve! Finding the Area to the Left of a Specified z-Score

36. Finding P(Z  1.23), that is, finding the area under the standard normal curve to the left of 1.23 Finding the Area to the Left of a Specified z-Score

37. Finding P(Z ≥ 0.76), that is, finding the area under the standard normal curve to the right of 0.76 Finding the Area to the Right of a Specified z-Score Always draw the curve!

38. Finding P(Z ≥ 0.76), that is, finding the area under the standard normal curve to the right of 0.76 Finding the Area to the Right of a Specified z-Score

39. Finding P(-0.68  Z  1.82), that is, finding the area under the standard normal curve that lies between –0.68 and 1.82 Find the Area Between Two Specified z-Scores Always draw the curve!

40. Finding P(-0.68  Z  1.82), that is, finding the area under the standard normal curve that lies between –0.68 and 1.82 Find the Area Between Two Specified z-Scores

41. Using Table II to find the area under the standard normal curve that lies (a) to the left of a specified z-score, (b) to the right of a specified z-score, (c) between two specified z-scores Summary

42. Finding the z-score having area 0.04 to its left, that is, finding z, so that P(Z  z)=0.04 Find the z-Score Having a Specified Area to Its Left Always draw the curve!

43. Inside the table, look for the best approximation to the given area z score Second decimal place for z-score

44. Finding the z-score having area 0.04 to its left, that is, finding z, so that P(Z  z)=0.04 Find the z-Score Having a Specified Area to Its Left

45. Useful Notation

46. Example-Find

48. Finding the z-Scores for a Specified Area Find the two z-scores that divide the area under the standard normal curve into a middle 0.95 area and two outside 0.025 areas, as depicted in the figure.

49. Finding the z-Scores for a Specified Area

50. Normal distribution calculations

51. Three nonstandard normal distributions

52. Standardizing three nonstandard normal distributions

53. The process of Standardizing preserves the areas in the sense indicated in above. Normal Distribution Calculations

54. and areas are probabilities, that is Normal Distribution Calculations

59. z-score for a z-score for b Normal Distribution Calculations

60. Percentage of people having IQs between 115 and 140

62. Conclusion: The percentage of people having IQs between 115 and 140 is 16.74% Percentage of people having IQs between 115 and 140

63. More Examples on Finding Probabilities

64. Example: The average on a statistics test was 78 with a standard deviation of 8. If the test scores are normally distributed, find the probability that a student receives a test score less than 90. Probability and Normal Distributions P(X < 90) = P(Z < 1.5) = 0.9332 The probability that a student receives a test score less than 90 is 0.9332. μ =0 z ? 1.5 90μ =78 P(X < 90) μ = 78 σ = 8 x

65. Example: The average on a statistics test was 78 with a standard deviation of 8. If the test scores are normally distributed, find the probability that a student receives a test score greater than than 85. Probability and Normal Distributions P(X > 85) = P(Z > 0.88) = 1  P(Z < 0.88) = 1  0.8106 = 0.1894 The probability that a student receives a test score greater than 85 is 0.1894. μ =0 z ? 0.88 85μ =78 P(X > 85) μ = 78 σ = 8 x

66. Example: The average on a statistics test was 78 with a standard deviation of 8. If the test scores are normally distributed, find the probability that a student receives a test score between 60 and 80. Probability and Normal Distributions P(60 < X < 80) = P(  2.25 < Z < 0.25) = P(Z < 0.25)  P(Z <  2.25) The probability that a student receives a test score between 60 and 80 is 0.5865. μ =0 z ?? 0.25  2.25 = 0.5987  0.0122 = 0.5865 6080μ =78 P(60 < X < 80) μ = 78 σ = 8 x

67. Normal Distributions: Finding Values

68. Finding a z-Score Given a Percentile Example: Find the z-score that corresponds to P 75. The z-score that corresponds to P 75 is the same z-score that corresponds to an area of 0.75. That is P 75 = 0.67 ?μ =0 z 0.67 Area = 0.75

69. To transform a standard z-score to a data value, x, in a given population, use the formula Example: The monthly electric bills in a city are normally distributed with a mean of $120 and a standard deviation of $16. Find the x-value corresponding to a z-score of 1.60. We can conclude that an electric bill of $145.60 is 1.6 standard deviations above the mean. Transforming a z-score to an x-score

70. Finding a Specific Data Value Example: The weights of bags of chips for a vending machine are normally distributed with a mean of 1.25 ounces and a standard deviation of 0.1 ounce. Bags that have weights in the lower 8% are too light and will not work in the machine. What is the least a bag of chips can weigh and still work in the machine? The least a bag can weigh and still work in the machine is 1.11 ounces. ? 0 z 8% P(Z < ?) = 0.08 P(Z <  1.41) = 0.08  1.41 1.25 x ? 1.11