Modeling Continuous Variables Lecture 24 Section 6.3.1 Fri, Feb 27, 2004

Models A mathematical model is an abstraction and, therefore, a simplification of a real situation. Real situations are usually much to complicated to deal with directly. For example, we might replace a jagged histogram with a smooth curve that has a simple equation.

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

Example of a Model Using a random number generator to simulate how a pair of rolled dice will land. Use the TI-83 and Enter randInt(2, 12), or Enter randInt(1, 6) twice and add the results. Which is right?

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

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

Histograms and Area In the histogram, we may replace the frequency with the proportion (of the total). Grade Frequency Proportion 60 – 69 3 0.12 70 – 79 8 0.32 80 – 89 9 0.36 90 – 99 5 0.20

Histograms and Area Proportion 0.40 0.30 0.20 0.10 Grade 60 70 80 90 60 70 80 90 100

Histograms and Area Proportion 0.40 0.30 0.20 0.10 Grade 60 70 80 90 60 70 80 90 100

Histograms and Area Furthermore, we may divide the proportions by the width of the classes to get the density. Grade Frequency Proportion Density 60 – 69 3 0.12 0.012 70 – 79 8 0.32 0.032 80 – 89 9 0.36 0.036 90 – 99 5 0.20 0.020

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

Histograms and Area The final 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 x 0.012) + (10 x 0.032) = 0.12 + 0.32 = 0.44 = 44%.

Density Functions This is the fundamental property that connects histograms with the graphs of continuous models that we use to approximate them, namely: The area under the curve between two numbers a and b on the x-axis represents the proportion of values that lie between a and b in the distribution.

Density Functions The area under the curve between a and b is the proportion. a b

Density Functions The area under the curve between a and b is the proportion. a b

Density Functions The area under the curve between a and b is the proportion. a b Area = Proportion

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

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

The Normal Distribution The normal distribution is the statistician’s name for the bell curve. It is a density function in the shape of a bell (sort of). Symmetric. Unimodal. Extends over the entire real line. Main part lies with 3 of the mean.

The Normal Distribution The curve has a bell shape (with infinitely long tails).

The Normal Distribution The mean  is located at the peak. 

The Normal Distribution The width of the “main” part of the curve is 6 standard deviations.  

The Normal Distribution The area under the entire curve is 1. Area = 1 

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).”

Various Normal Distributions 1 2 3 4 5 6 7 8

Various Normal Distributions 1 2 3 4 5 6 7 8

Various Normal Distributions 1 2 3 4 5 6 7 8

Various Normal Distributions 1 2 3 4 5 6 7 8