1. Modeling Continuous Variables Lecture 19 Section 6.1 - 6.3.1 Fri, Feb 24, 2006

2. Models Mathematical model – An abstraction and, therefore, a simplification of a real situation, one that retains the essential features. Mathematical model – An abstraction and, therefore, a simplification of a real situation, one that retains the essential features. Real situations are usually much to complicated to deal with in all their details. Real situations are usually much to complicated to deal with in all their details.

3. Examples Economic models treat money as a continuous quantity, even though it is discrete. Economic models treat money as a continuous quantity, even though it is discrete. This is an abstraction that is incorporated into the model to make it simpler. This is an abstraction that is incorporated into the model to make it simpler. The “bell curve” is a model (an abstraction) of many populations. The “bell curve” is a model (an abstraction) of many populations. Real populations have all sorts of bumps and twists. Real populations have all sorts of bumps and twists. The bell curve is smooth and perfectly symmetric. The bell curve is smooth and perfectly symmetric.

4. Models No mathematical model is perfect. No mathematical model is perfect. A mathematical model is useful (and powerful) to the extent that it is a faithful representation of reality. A mathematical model is useful (and powerful) to the extent that it is a faithful representation of reality. Conversely, to the extent that is it not faithful to reality, it can lead to false conclusions about the situation that it is supposed to model. Conversely, to the extent that is it not faithful to reality, it can lead to false conclusions about the situation that it is supposed to model.

5. Example of a Model Use a random number generator to simulate how a pair of rolled dice will land. Use a random number generator to simulate how a pair of rolled dice will land. The possible totals range from 2 to 12. The possible totals range from 2 to 12. Using the TI-83, which is a correct model? Using the TI-83, which is a correct model? Enter randInt(2, 12), that is, get a random number from 2 to 12, or Enter randInt(2, 12), that is, get a random number from 2 to 12, or Enter 2*randInt(1, 6), that is, double a random number from 1 to 6, or Enter 2*randInt(1, 6), that is, double a random number from 1 to 6, or Enter randInt(1, 6) + randInt(1, 6), that is, add two random numbers from 1 to 6. Enter randInt(1, 6) + randInt(1, 6), that is, add two random numbers from 1 to 6.

6. Histograms and Area If a histogram is drawn appropriately, then frequency is represented by area. If a histogram is drawn appropriately, then frequency is represented by area. Consider the following histogram of test scores. Consider the following histogram of test scores. GradeFrequency 60 – 69 3 70 – 79 8 80 – 89 9 90 – 99 5

7. Histograms and Area Grade Frequency 60 70 8090 100 0 2 4 6 8 10

8. Histograms and Area In the histogram, we may replace the frequency with the proportion (of the total). In the histogram, we may replace the frequency with the proportion (of the total). GradeFrequencyProportion 60 – 69 30.12 70 – 79 80.32 80 – 89 90.36 90 – 99 50.20

9. Histograms and Area Grade Proportion 60 70 8090 100 0 0.10 0.20 0.30 0.40

10. Histograms and Area Grade Proportion 60 70 8090 100 0 0.10 0.20 0.30 0.40

11. Histograms and Area Furthermore, we may divide the proportions by the width of the classes to get the density. Furthermore, we may divide the proportions by the width of the classes to get the density. GradeFrequencyProportionDensity 60 – 69 30.120.012 70 – 79 80.320.032 80 – 89 90.360.036 90 – 99 50.200.020

12. Histograms and Area Grade Density 60 70 8090 100 0 0.010 0.020 0.030 0.040

13. Histograms and Area The final histogram has the special property that the proportion can be found by computing the area of the rectangle. The final histogram has the special property that the proportion can be found by computing the area of the rectangle. The vertical scale has been adjusted so that the total area is 1, or 100%. The vertical scale has been adjusted so that the total area is 1, or 100%. For example, what proportion of the grades are less than 80? For example, what proportion of the grades are less than 80? Compute: (10  0.012) + (10  0.032) Compute: (10  0.012) + (10  0.032) = 0.12 + 0.32 = 0.44 = 44%.

14. Density Functions This is the fundamental property that connects the graph of a continuous model to the population that it represents, namely: This is the fundamental property that connects the graph of a continuous model to the population that it represents, namely: The area under the graph between two numbers a and b on the x-axis represents the proportion of the population that lies between a and b. The area under the graph between two numbers a and b on the x-axis represents the proportion of the population that lies between a and b. AREA = PROPORTION

15. Density Functions The area under the curve between a and b is the proportion of the values of x that lie between a and b. The area under the curve between a and b is the proportion of the values of x that lie between a and b. ab x

16. Density Functions The area under the curve between a and b is the proportion of the values of x that lie between a and b. The area under the curve between a and b is the proportion of the values of x that lie between a and b. ab x

17. Density Functions The area under the curve between a and b is the proportion of the values of x that lie between a and b. The area under the curve between a and b is the proportion of the values of x that lie between a and b. ab Area = Proportion x

18. Density Functions A consequence of this is that the total area under the curve must be 1, representing a proportion of 100%. A consequence of this is that the total area under the curve must be 1, representing a proportion of 100%. ab x

19. Density Functions A consequence of this is that the total area under the curve must be 1, representing a proportion of 100%. A consequence of this is that the total area under the curve must be 1, representing a proportion of 100%. ab 100% x

20. The Normal Distribution Normal distribution – The statistician’s name for the bell curve. Normal distribution – The statistician’s name for the bell curve. It is a density function in the shape of a bell (sort of). It is a density function in the shape of a bell (sort of). Symmetric. Symmetric. Unimodal. Unimodal. Extends over the entire real line (no endpoints). Extends over the entire real line (no endpoints). “Main part” lies within  3  of the mean. “Main part” lies within  3  of the mean.

21. The Normal Distribution The curve has a bell shape, with infinitely long tails in both directions. The curve has a bell shape, with infinitely long tails in both directions.

22. The Normal Distribution The mean  is located in the center, at the peak. The mean  is located in the center, at the peak. 

23. The Normal Distribution The width of the “main” part of the curve is 6 standard deviations wide (3 standard deviations each way from the mean). The width of the “main” part of the curve is 6 standard deviations wide (3 standard deviations each way from the mean).   – 3   + 3 

24. The Normal Distribution The area under the entire curve is 1. The area under the entire curve is 1. (The area outside of  3 st. dev. is approx. 0.0027.) (The area outside of  3 st. dev. is approx. 0.0027.) Area = 1  – 3   + 3 

25. The Normal Distribution The normal distribution with mean  and standard deviation  is denoted N( ,  ). The normal distribution with mean  and standard deviation  is denoted N( ,  ). For example, if X is a variable whose distribution is normal with mean 30 and standard deviation 5, then we say that “X is N(30, 5).” For example, if X is a variable whose distribution is normal with mean 30 and standard deviation 5, then we say that “X is N(30, 5).”

26. The Normal Distribution If X is N(30, 5), then the distribution of X looks like this: If X is N(30, 5), then the distribution of X looks like this: 301545

27. Some Normal Distributions 0 1 2 3 4 5 6 87 N(3, 1)

28. Some Normal Distributions 0 1 2 3 4 5 6 87 N(3, 1) N(5, 1)

29. Some Normal Distributions 0 1 2 3 4 5 6 87 N(3, 1) N(2, ½) N(5, 1)

30. Some Normal Distributions 0 1 2 3 4 5 6 87 N(3, 1) N(2, ½) N(5, 1) N(3½, 1½)

31. Bag A vs. Bag B Suppose we have two bags, Bag A and Bag B. Suppose we have two bags, Bag A and Bag B. Each bag contains millions of vouchers. Each bag contains millions of vouchers. In Bag A, the values of the vouchers have distribution N(50, 10). In Bag A, the values of the vouchers have distribution N(50, 10). Normal with  = $50 and  = $10. Normal with  = $50 and  = $10. In Bag B, the values of the vouchers have distribution N(80, 15). In Bag B, the values of the vouchers have distribution N(80, 15). Normal with  = $80 and  = $15. Normal with  = $80 and  = $15.

32. Bag A vs. Bag B 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B

33. Bag A vs. Bag B We are presented with one of the bags. We are presented with one of the bags. We select one voucher at random from that bag. We select one voucher at random from that bag. 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B

34. Bag A vs. Bag B If its value is less than or equal to $65, then we will decide that it was from Bag A. If its value is less than or equal to $65, then we will decide that it was from Bag A. 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65

35. Bag A vs. Bag B If its value is less than or equal to $65, then we will decide that it was from Bag A. If its value is less than or equal to $65, then we will decide that it was from Bag A. 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65 Acceptance Region

36. Bag A vs. Bag B If its value is less than or equal to $65, then we will decide that it was from Bag A. If its value is less than or equal to $65, then we will decide that it was from Bag A. 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65 Acceptance RegionRejection Region

37. Bag A vs. Bag B What is  ? What is  ? 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65

38. Bag A vs. Bag B What is  ? What is  ? 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65 

39. Bag A vs. Bag B What is  ? What is  ? 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65

40. Bag A vs. Bag B What is  ? What is  ? 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65 

41. Bag A vs. Bag B If the distributions are very close together, then  and  will be large. If the distributions are very close together, then  and  will be large. 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65

42. Bag A vs. Bag B If the distributions are very similar, then  and  will be large. If the distributions are very similar, then  and  will be large. 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65 

43. Bag A vs. Bag B If the distributions are very similar, then  and  will be large. If the distributions are very similar, then  and  will be large. 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65 

44. Bag A vs. Bag B Similarly, if the distributions are far apart, then  and  will both be very small. Similarly, if the distributions are far apart, then  and  will both be very small.

45. Bag A vs. Bag B What is the p-value of $75? What is the p-value of $75? 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65 75

46. Bag A vs. Bag B What is the p-value of $75? What is the p-value of $75? 30 40 50 60 70 9080 100110 H 0 : Bag A H 1 : Bag B 65 75 p-value