Unit 10 notes Fall23

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BUSN 3000 Unit 10: Simple Linear Regression Unit 10: Simple Linear Regression Linear regression A linear regression model describes how a response variable changes as an explanatory variable changes. ^ y = b 0 + b 1 ( x ) In JMP: Analyze – Fit Y by X Interpreting coefficients The slope is the predicted change in Y for a 1-unit increase in X . The intercept is the predicted value of Y when X = 0. Extrapolation occurs when we try to make a prediction based on an X value that is outside the range of our data. These predictions may not be accurate. Making predictions and calculating residuals Predict the price for an 8-year-old car listed on Craigslist. The error between the actual value y and the predicted value ^ y is called the residual : e = y ^ y o Positive residual o Negative residual Calculate the residual for an 8-year-old car that’s listed for $34,995. 1
BUSN 3000 Unit 10: Simple Linear Regression How strong is the association? The correlation ( r ) measures how closely X and Y follow a linear relationship (how close the points are to a straight line). Use the Guess the Correlation applet to try it out. o If the points fall in an almost perfect, negative linear pattern, r is close to _______. o If the points fall in an almost perfect, positive linear pattern, r is close to _______. o If there is almost no linear relationship, r is close to _______. o Is correlation resistant to outliers? The coefficient of determination, R 2 , tells us the proportion of variation in Y that can be predicted (explained) by the model. o If the R 2 value is close to _______, the regression line provides very accurate predictions. o If the R 2 value is close to _______, the regression line is not very useful in making predictions. o Interpret R 2 in context for the used cars example. o Calculate the correlation r for the used cars example. The root mean square error (RMSE) is the average size of the prediction errors (residuals). o Interpret RMSE in context for the used cars example. Consider changes to the dataset What would happen if we changed the units of age from years to months? o The slope would… o The R-sq and correlation values would… o The RMSE value would… Suppose someone listed a 1-year-old car for $0. What would happen if this point were added to the dataset? o The slope would… o The R-sq and correlation values would… o The RMSE value would… 2
BUSN 3000 Unit 10: Simple Linear Regression Inference for Regression The population and the sample Earlier, we built a model to predict a response variable y by fitting a straight line to a sample data set: ^ y = b 0 + b 1 ( x ) We want to draw inferences about the true relationship between x and y in the population . Hypothesis tests Does a linear relationship really exist or is it plausible that the sample slope occurred just by chance? H 0 : H A : Reasoning about strength of evidence: Which graph provides the strongest evidence against H 0 ? The weakest? Test statistics and p-values test stat = sample statistic nullhypothesis value standard error →t = b 1 S E b 1 How large is large enough? o Compare to a t distribution with df = n ¿ of predictors 1 3
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