invest_3ed.pdf

# Statistics 285 cm s x 345 cm y 6775 in and s y 500 in

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statistics: = 28.5 cm, s x = 3.45 cm, y = 67.75 in and s y = 5.00 in, with r = 0.711. Use these statistics and the formulas above to calculate the coefficients of the least-squares regression line. Confirm that these agree with what the applet reported for the equation of the least squares line. (w) Use this least-squares regression line to predict the height of a person with a 28 cm foot length. Then repeat for a person with a 29-cm foot length. Calculate the difference in these two height predictions. Does this value look familiar? Explain. (x) Provide an interpretation of the slope coefficient ( b 1 ) in this context. (y) Provide an interpretation of the intercept coefficient ( b 0 ) in this context. Is such an interpretation meaningful for these data? Explain. x x

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Chance/Rossman, 2015 ISCAM III Investigation 5.8 372 Discussion: The slope coefficient of 1.03 indicates that the predicted height of a person increases by 1.03 inches for each additional centimeter of foot length. In other words, i f one person’s foot length is one cm longer than another’s, we predict this person to be 1.03 inches taller than the other person. Notice we are being careful to talk about the “ predicted ” change or “ average ” change, because these are estimates based on the least squares line, not an exact mathematical relationship between height and foot length. The intercept coefficient can be interpreted as the predicted height for a person whose foot length is zero not a very sensible prediction in this context. In fact, the intercept will often be too far outside the range of x values for us to seriously consider its interpretation. (z) Use the least squares regression line to predict the height of someone whose foot length is 44 cm. Explain why you should not be as comfortable making this prediction as the ones in (w). Definition: Extrapolation refers to making predictions outside the range of the explanatory variable values in the data set used to determine the regression line. It is generally ill-advised. Evaluating the model One way to assess the usefulness of this least-squares line is to measure the improvement in our predictions by using the least-squares line instead of the y line that assumes no knowledge about the explanatory variable. (aa) In the applet, uncheck and recheck the Show Movable Line box. Check the Show Squared Residuals box to determine the SSE if we were to use y as our predicted value for every x (foot size). Record this value. SSE ( y ) = (bb) Recall the SSE value for the regression line. Determine the percentage change in the SSE between the y line and the regression line: 100% u [( SSE ( y ) ± SSE (least-squares))/ SSE ( y )] = Definition: The preceding expression indicates the reduction in the prediction errors from using the least squares line instead of the y line. This is referred to as the coefficient of determination , interpreted as the percentage of the variability in the response variable that is explained by the least-squares regression line with the explanatory variable. This provides us with a measure of how accurate our predictions will be and is most useful for comparing different models (e.g., different choices of explanatory variable).
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