Inference for Refression (Essay Sample)

Week 11 Overview

Jul 27, 2015 9:39 AM

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Welcome to session 11. This week, we will describe the relationship and possible cause/effect among two variables. For example, if a student studies more, that student may possibly do better on their SAT exam. We know that higher SAT scores correspond with better GPAs in college; however, do higher SAT scores cause higher grades? This week, we will introduce the idea of scatterplots, which give us a visual sense of the relationship between two variables. We will also look at the correlation coefficient, which gives us a quantitative measure of the strength and direction of a linear relationship. Finally, we will look at linear regression which assumes cause/effect and produces a straight-line approximation for a set of paired variables. We will look at paired sets of numbers and if they move together, they are correlated. If one quantity increases and the other quantity tends to increase, the correlation is positive (and if one variable increases while the other decreases, the correlation is negative). The correlation coefficient is denoted by r (the Greek letter rho) with a range of between -1 and 1. If the number of standard deviations from the mean of one variable corresponds exactly to the number for other variable, then r = 1. Measuring correlations can reveal some surprising associations. Consider the data set {(1,1), (2,2), (3,3), (4,4), (5,5), (0,7)}. The correlation of this data set is zero, even though 5 out of 6 of the data pairs are perfectly correlated. In the case of a perfect positive correlation (r=1), our scatterplot will show all the data points on a straight line with positive slope. This straight line is a kind of summary of the data and is called a linear regres...