chapter11_notes3 - Notes Introduction to Probability and...

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Introduction to Probability and Statistics II Instructor: John Snyder Office: Middlebush 35 Office hours: MW 8am - 9:30 pm Email: [email protected] 2 18 2016 John Snyder (MU) Stat —— 3500 — 2 18 2016 1 / 67 Simple Linear Regression 1 11.1 - Probabilistic Models 2 11.2 - Least Squares Estimation 3 11.3 - Model Assumptions 4 11.4 - Inference about β 1 5 11.5 - The Coefficients of Correlation and Determination 6 11.6 - Using the Model for Estimation and Prediction John Snyder (MU) Stat —— 3500 — 2 18 2016 2 / 67 11.1 - Probabilistic Models Section 1 11.1 - Probabilistic Models John Snyder (MU) Stat —— 3500 — 2 18 2016 3 / 67 Notes Notes Notes
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11.1 - Probabilistic Models Introduction Chapter 9 described methods of handling data whose independent variables are qualitative . Chapter 11(and 12) will tell describe methods of analyzing data whose independent variables are quantitative . For example: Say a response variable is the height of a person. In chapter 9 we would consider factors like Gender or Ethnicity. In chapter 11 we will consider factors like age. Instead of the independent variable (or factor) having a discrete number of values, now it can have infinitely many values. John Snyder (MU) Stat —— 3500 — 2 18 2016 4 / 67 11.1 - Probabilistic Models What is a Model? A model is a mathematical equation which relates a dependent (response) variable to one or more independent variables. Exact relationships are represented with a deterministic model. A = π r 2 0 1 2 3 4 5 0 20 40 60 80 r A Here the dependent variable is A , the area of a circle, you can exactly determine the value of A just by knowing r , the radius of the circle. John Snyder (MU) Stat —— 3500 — 2 18 2016 5 / 67 11.1 - Probabilistic Models Probabilistic Models In statistics, we consider approximating models in the presence of random error . These are called probabilistic models. Y = π x 2 + ε 0 1 2 3 4 0 10 20 30 40 50 x Y Here the dependent variable is Y , but in this case, we cannot exactly determine the value of Y if we know the value of x . This is because there is random error( ε ) that must be factored in. John Snyder (MU) Stat —— 3500 — 2 18 2016 6 / 67 Notes Notes Notes
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11.1 - Probabilistic Models Probabilistic Models For example, say we are studying the relationship between weight and height. We might use a model like Weight = 2 . 5 · Height + ε Clearly, not everyone who is 6 tall will weigh 180 pounds. But, we do know there is some relationship between height and weight. We use probabilistic models when we know there will be unexplained variation in our response variable that is not accounted for with our factor(s). In practice, we will not have the value for the slope(2.5 here) and will seek to estimate that effect. John Snyder (MU) Stat —— 3500 — 2 18 2016 7 / 67 11.1 - Probabilistic Models Probabilistic Models The general form for the probabilistic models we will discuss is y = deterministic component + random error The deterministic part of the relationship is the part that describes the ”line.” The probabilistic part of the relationship is the part that tells us that the relationship is not exact.
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