UCLA STATS 101A - stats 101A notes (92 pages)

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Learning Objectives After careful study of this chapter you should be able to do the following 1 Use simple linear regression for building empirical models to engineering and scientific data 2 Understand how the method of least squares is used to estimate the parameters in a linear regression model 3 Analyze residuals to determine if the regression model is an adequate fit to the data or to see if any underlying assumptions are violated 4 Test the statistical hypotheses and construct confidence intervals on the regression model parameters 5 Use the regression model to make a prediction of a future observation and construct an appropriate prediction interval on the future observation 6 Apply the correlation model 7 Use simple transformations to achieve a linear regression model 1 Spurious Correlations 2 3 4 5 Empirical Models Many problems in engineering and science involve exploring the relationships between two or more variables Regression analysis is a statistical technique that is very useful for these types of problems For example in a chemical process suppose that the yield of the product is related to the process operating temperature Regression analysis can be used to build a model to predict yield at a given temperature level 6 Empirical Models 7 Empirical Models 8 Empirical Models Based on the scatter diagram it is probably reasonable to assume that the mean of the random variable Y is related to x by the following straight line relationship where the slope and intercept of the line are called regression coefficients The simple linear regression model is given by where is the random error term 9 Empirical Models We think of the regression model as an empirical model Suppose that the mean and variance of are 0 and 2 respectively then The variance of Y given x is 10 Empirical Models The true regression model is a line of mean values where 1 can be interpreted as the change in the mean of Y for a unit change in x Also the variability of Y at a particular value