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Examples of continuous probability distributions:

Examples of continuous probability distributions:. The normal and standard normal. The Normal Distribution. f(X). Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread. σ. μ. X.

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Examples of continuous probability distributions:

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  1. Examples of continuous probability distributions: The normal and standard normal

  2. The Normal Distribution f(X) Changingμshifts the distribution left or right. Changing σ increases or decreases the spread. σ μ X

  3. This is a bell shaped curve with different centers and spreads depending on  and  The Normal Distribution:as mathematical function (pdf) Note constants: =3.14159 e=2.71828

  4. The Normal PDF It’s a probability function, so no matter what the values of  and , must integrate to 1!

  5. Normal distribution is defined by its mean and standard dev. E(X)= = Var(X)=2 = Standard Deviation(X)=

  6. **The beauty of the normal curve: 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.

  7. 68% of the data 95% of the data 99.7% of the data 68-95-99.7 Rule

  8. 68-95-99.7 Rulein Math terms…

  9. How good is rule for real data? Check some example data: The mean of the weight of the women = 127.8 The standard deviation (SD) = 15.5

  10. 112.3 143.3 68% of 120 = .68x120 = ~ 82 runners In fact, 79 runners fall within 1-SD (15.5 lbs) of the mean. 127.8

  11. 96.8 158.8 95% of 120 = .95 x 120 = ~ 114 runners In fact, 115 runners fall within 2-SD’s of the mean. 127.8

  12. 81.3 174.3 99.7% of 120 = .997 x 120 = 119.6 runners In fact, all 120 runners fall within 3-SD’s of the mean. 127.8

  13. Example • Suppose SAT scores roughly follows a normal distribution in the U.S. population of college-bound students (with range restricted to 200-800), and the average math SAT is 500 with a standard deviation of 50, then: • 68% of students will have scores between 450 and 550 • 95% will be between 400 and 600 • 99.7% will be between 350 and 650

  14. Example • BUT… • What if you wanted to know the math SAT score corresponding to the 90th percentile (=90% of students are lower)? P(X≤Q) = .90  Solve for Q?….Yikes!

  15. The Standard Normal (Z):“Universal Currency” The formula for the standardized normal probability density function is

  16. The Standard Normal Distribution (Z) All normal distributions can be converted into the standard normal curve by subtracting the mean and dividing by the standard deviation: Somebody calculated all the integrals for the standard normal and put them in a table! So we never have to integrate! Even better, computers now do all the integration.

  17. Comparing X and Z units 100 200 X ( = 100,  = 50) 0 2.0 Z ( = 0,  = 1)

  18. Example • For example: What’s the probability of getting a math SAT score of 575 or less, =500 and =50? • i.e., A score of 575 is 1.5 standard deviations above the mean Yikes! But to look up Z= 1.5 in standard normal chart (or enter into SAS) no problem! = .9332

  19. Z=1.50 Z=1.50 Looking up probabilities in the standard normal table What is the area to the left of Z=1.50 in a standard normal curve? Area is 93.32%

  20. Z=1.51 Z=1.51 Looking up probabilities in the standard normal table What is the area to the left of Z=1.51 in a standard normal curve? Area is 93.45%

  21. Probit function: the inverse (area)= Z: gives the Z-value that goes with the probability you want  For example, recall SAT math scores example. What’s the score that corresponds to the 90th percentile? In the Table, find the Z-value that corresponds to an area of 90%...

  22. 90% area corresponds to a Z score of about 1.28.

  23. Probit function: the inverse Z=1.28; convert back to raw SAT score  1.28 = X – 500 =1.28 (50) X=1.28(50) + 500 = 564 (1.28 standard deviations above the mean!)

  24. Practice problem If birth weights in a population are normally distributed with a mean of 109 oz and a standard deviation of 13 oz, • What is the chance of obtaining a birth weight of 141 oz or heavier when sampling birth records at random? • What is the chance of obtaining a birth weight of 120 or lighter?

  25. Answer • What is the chance of obtaining a birth weight of 141 oz or heavier when sampling birth records at random?

  26. Area to the left of Z=2.46 is .9931 Area to the right of 2.46 is: 1-.9931 = .0069 or .69%

  27. Answer b. What is the chance of obtaining a birth weight of 120 or lighter?

  28. Area to the left of Z=0.85 is .8023 or 80.23%.

  29. Practice problem 2: DSST (a measure of cognitive function) is a normally distributed trait… Normally distributed Mean = 28 points Standard deviation = 10 points

  30. Practice problem 2 • a. What percent of people have values of DSST above 38? • b. What percent of people have values of DSST below 8?

  31. Answers • a. What percent of people have values of DSST above 38? Thus, 16% of people have DSST’s above 38.

  32. Answers • b. What percent of people have values of DSST below 8? Thus, 2.5% of people have DSST’s below 8.

  33. Review question 1 The probability that a standardized normal variable Z is positive is ____. • 100% • 50% • 10% • 0%

  34. Review question 2 The probability that Z is between -2 and -1 is _____. • 50% • 34% • 25.5% • 13.5%

  35. Review question 3 The probability that Z values are larger than _____ is 0.6985. • Z=1 • Z=0 • Z=-.5 • Z=+.5

  36. Review question 4 27% of Z values are smaller than ____. • Z=0 • Z=1 • Z=-.6 • Z=+.6

  37. Are my data “normal”? • Not all continuous random variables are normally distributed!! • It is important to evaluate how well the data are approximated by a normal distribution

  38. Are my data normally distributed? • Look at the histogram! Does it appear bell shaped? • Compute descriptive summary measures—are mean, median, and mode similar? • Do 2/3 of observations lie within 1 std dev of the mean? Do 95% of observations lie within 2 std dev of the mean? • Look at a normal probability plot—is it approximately linear? • Run tests of normality (such as Kolmogorov-Smirnov). But, be cautious, highly influenced by sample size!

  39. Data from our class… Median = 8 Mean = 8.8 Mode = 0 SD = 8.3 Range = 0 to 32 (= 4 σ)

  40. Data from our class… Median = 45 Mean = 41 Mode = 6 SD = 23 Range = 0 to 83 (~ 3.5 σ)

  41. Data from our class… Median = 4 Mean = 3.7 Mode = 4 SD = 1.8 Range = 0.5 to 7 (~ 3.5 σ)

  42. Data from our class… Median = 18 Mean = 20 Mode = 20 SD = 16 Range = 2 to 70 (~4 σ)

  43. 17.1 0.5 Data from our class… 8.8 +/- 8.3 = 0.5 – 17.1

  44. Data from our class… 8.8 +/- 2*8.3 = 0 – 25.4

  45. Data from our class… 8.8 +/- 3*8.3 = 0 – 33.7

  46. 18 64 Data from our class… 41 +/- 23 = 18 – 64

  47. 87 0 Data from our class… 41 +/- 2*23 = 0 – 87

  48. 100 0 Data from our class… 41 +/- 3*23 = 0– 100

  49. 1.9 5.5 Data from our class… 3.7 +/- 1.8= 1.9 – 5.5

  50. 0.1 7.3 Data from our class… 3.7 +/- 2*1.8= 0.1 – 7.3

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