Modeling Continuous Variables Lecture 19 Section Mon, Oct 11, 2004

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. For example, we often treat money as a continuous quantity, even though it is discrete. For example, we often treat money as a continuous quantity, even though it is discrete.

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.

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), or Enter randInt(2, 12), or Enter randInt(1, 6) twice and add the results, that is, enter randInt(1, 6) + randInt(1, 6). Enter randInt(1, 6) twice and add the results, that is, enter randInt(1, 6) + randInt(1, 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 a histogram of the following test scores. Consider a histogram of the following test scores. GradeFrequency 60 – – – – 99 5

Histograms and Area Grade Frequency

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 – – – –

Histograms and Area Grade Proportion

Histograms and Area Grade Proportion

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 – – – –

Histograms and Area Grade Density

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. 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) = = 0.44 = 44%.

Density Functions This is the fundamental property that connects histograms with the graphs of continuous models that we use to approximate them, namely: 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. 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. The area under the curve between a and b is the proportion. ab

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

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

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%

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. Extends over the entire real line. Main part lies within  3  of the mean. Main part lies within  3  of the mean.

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

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

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 

The Normal Distribution The area under the entire curve is 1. The area under the entire curve is 1. Area = 1  – 3   + 3 

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).”

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:

Various Normal Distributions N(3, 1)

Various Normal Distributions N(3, 1) N(5, 1)

Various Normal Distributions N(3, 1) N(2, ½) N(5, 1)

Various Normal Distributions N(3, 1) N(2, ½) N(5, 1) N(3½, 1½)