1. Modeling Continuous Variables
Lecture 19
Section
Fri, Oct 6, 2006

2. Models
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.

3. Example
The “bell curve” is a model (an abstraction) of many populations.
Real populations have all sorts of bumps and twists and irregularities.
The bell curve is smooth and perfectly symmetric.
In statistics, the bell curve is called the normal curve, or normal distribution.

4. Models
Our models will be models of distributions, presented either as histograms or as continuous distributions.

5. Histograms and Area In a histogram, frequency is represented by area.
Consider the following distribution of test scores.
Grade
Frequency
60 – 69 3
70 – 79 8
80 – 89 9
90 – 99 5

6. Histograms and Area
Frequency 10 8 6 4 2
Grade 60 70 80 90
100

7. Histograms and Area What is the total area of this histogram?
We will rescale the vertical scale so that the total area equals 1, representing 100%.

8. Histograms and Area
To achieve this, we divide the frequencies by the original area to get the density.
Grade
Frequency
Density
60 – 69 3
0.012
70 – 79 8
0.032
80 – 89 9
0.036
90 – 99 5
0.020

9. Histograms and Area Density 0.040 0.030 0.020 0.010 Grade 60 70 80 9060 70 80 90
100

10. Histograms and Area Density 0.040 Total area = 1 0.030 0.020 0.010
Grade 60 70 80 90
100

11. Histograms and Area
This histogram has the special property that the proportion can be found by computing the area of the rectangle.
For example, what proportion of the grades are less than 80?
Compute: (10  0.012) + (10  0.032)
= = 0.44 = 44%.

12. Density Functions
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.
AREA = PROPORTION

13. Density Functions Now consider an arbitrary distribution.
The area under the curve between a and b is the proportion of the values of x that lie between a and b. x a b

14. Density Functions Now consider an arbitrary distribution.
The area under the curve between a and b is the proportion of the values of x that lie between a and b. x a b

15. Density Functions Now consider an arbitrary distribution.
The area under the curve between a and b is the proportion of the values of x that lie between a and b. x a b
Area = Proportion

16. Density Functions
Again, the total area under the curve must be 1, representing a proportion of 100%. x a b

17. Density Functions
Again, the total area under the curve must be 1, representing a proportion of 100%.
100% x a b

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

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

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

21. 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). 
 – 3 
 + 3

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

23. The Normal Distribution
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).”

24. The Normal Distribution
If X is N(30, 5), then the distribution of X looks like this: 15 30 45

25. Some Normal Distributions1 2 3 4 5 6 7 8

26. Some Normal Distributions1 2 3 4 5 6 7 8

27. Some Normal Distributions1 2 3 4 5 6 7 8

28. Some Normal Distributions1 2 3 4 5 6 7 8

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

30. Bag A vs. Bag B
H0: Bag A
H1: Bag B 30 40 50 60 70 80 90
100
110

31. Bag A vs. Bag B We are presented with one of the bags.
We select one voucher at random from that bag.
H0: Bag A
H1: Bag B 30 40 50 60 70 80 90
100
110

32. 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.
H0: Bag A
H1: Bag B 30 40 50 60 65 70 80 90
100
110

33. 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.
H0: Bag A
H1: Bag B 30 40 50 60 65 70 80 90
100
110
Acceptance Region

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.
H0: Bag A
H1: Bag B 30 40 50 60 65 70 80 90
100
110
Acceptance Region
Rejection Region

35. Bag A vs. Bag B What is ? H0: Bag A H1: Bag B 30 40 50 60 65 70 80 90
100
110

36. Bag A vs. Bag B What is ? H0: Bag A H1: Bag B  30 40 50 60 65 70 8090
100
110

37. Bag A vs. Bag B What is ? H0: Bag A H1: Bag B 30 40 50 60 65 70 80 90
100
110

38. Bag A vs. Bag B What is ? H0: Bag A H1: Bag B  30 40 50 60 65 70 8090
100
110

39. Bag A vs. Bag B
If the distributions are very close together, then  and  will be large.
H0: Bag A
H1: Bag B 30 40 50 60 65 70 80 90
100
110

40. Bag A vs. Bag B
If the distributions are very similar, then  and  will be large.
H0: Bag A
H1: Bag B  30 40 50 60 65 70 80 90
100
110

41. Bag A vs. Bag B
If the distributions are very similar, then  and  will be large.
H0: Bag A
H1: Bag B  30 40 50 60 65 70 80 90
100
110

42. Bag A vs. Bag B
Similarly, if the distributions are far apart, then  and  will both be very small.
H0: Bag A
H1: Bag B 30 40 50 60 65 70 80 90
100
110

43. Bag A vs. Bag B
Similarly, if the distributions are far apart, then  and  will both be very small.
H0: Bag A
H1: Bag B  30 40 50 60 65 70 80 90
100
110

44. Bag A vs. Bag B
Similarly, if the distributions are far apart, then  and  will both be very small.
H0: Bag A
H1: Bag B  30 40 50 60 65 70 80 90
100
110