Exam 13: Inference in Linear Models

Use the given set of points to compute the residual standard deviation . x 12 20 19 17 19 10 y 43 69 65 58 64 36 x=13

Use the given set of points to construct a $95 \%$ prediction interval for an individual response for the given value of $x$ .

The following MINITAB output presents a multiple regression equation $y = b _ { 0 } + b _ { 1 } x _ { 1 } + b _ { 2 } x _ { 2 } + b _ { 3 } x _ { 3 }$ $+ b _ { 4 } x _ { 4 }$ . The regression equation is $\mathrm { Y } = 3.9695 + 1.4577 \mathrm { X } 1 - 1.7859 \mathrm { X } 2 + 0.7686 \mathrm { X } 3 + 0.0777 \mathrm { X } 4$ Predictor Coef SE Coef T P Constant 3.9695 0.8785 0.9299 0.327 X1 1.4577 0.6034 3.5107 0.003 X2 -1.7859 0.7302 -3.1148 0.005 X3 0.7686 0.6732 1.9294 0.088 X4 0.0777 0.7569 -1.0782 0.352 $\text { Analysis of Variance }$ Source DF SS MS F P Regression 4 1,148.7 287.2 9.0031 0.003 Residual Error 34 1,083.9 31.9 Total 38 2,232.6 Is the model useful for prediction? Use the  = 0.05 level.

In a study of reaction times, the time to respond to a visual stimulus (x) and the time to respond to an auditory stimulus (y) were recorded for each of 8 subjects. Times were measured in thousandths Of a second. The results are presented in the following table. Visual Auditory 193 189 240 222 239 226 236 225 200 189 161 158 214 209 204 193 Construct a 95% confidence interval for the slope of the least-squares regression line.

Use the given set of points to test the null hypothesis $H _ { 0 } : \beta _ { 1 } = 0 \text { versus } H _ { 1 } : \beta _ { 1 } \neq 0 . \text { Use the } \alpha = 0.01$ level of significance. x 14 7 15 10 9 6 14 15 y 32 17 36 26 21 15 34 37

The following MINITAB output presents a confidence interval for a mean response and a prediction interval for an individual response. New Obs Fit SE Fit 95.0\% CI 95.0\% PI 1 9.582 1.189 (3.264,15.900) (2.795,16.369) Values of Predictors for New Observations New Obs X1 X2 X3 1 1.94 1.11 1.17 What is the 95% confidence interval for the mean response?

In a study of reaction times, the time to respond to a visual stimulus (x) and the time to respond to an auditory stimulus (y) were recorded for each of 8 subjects. Times were measured in thousandths of A second. The results are presented in the following table. Visual Auditory 249 246 194 241 210 244 197 245 208 243 166 243 239 252 151 242 Construct a 95% prediction interval for the auditory response time for a particular subject whose Visual response time is 182.

Use the given set of points to compute . x 11 5 7 6 9 8 14 12 y 24 17 19 20 23 17 30 25

The following MINITAB output presents a 95% confidence interval for the mean ozone level on days when the relative humidity is 40%, and a 95% prediction interval for the ozone level on a Particular day when the relative humidity is 40%. The units of ozone are parts per billion. Predicted Values for New Observations New Obs Fit SE Fit 95.0\% CI 95.0\% PI 1 44.17 1.5 (41.23,47.11) (28.00,60.34) Values of Predictors for New Observations What is the $95 \%$ confidence interval for the mean ozone level for days when the relative humidity is $40 \%$ ?

Use the given set of points to compute the sum of squares for x,(x- . x 5 9 10 7 7 10 6 8 y 16 22 27 20 21 22 20 21

In a study of reaction times, the time to respond to a visual stimulus (x) and the time to respond to an auditory stimulus (y) were recorded for each of 8 subjects. Times were measured in thousandths of a second. The results are presented in the following table. Visual Auditory 168 245 171 242 185 240 235 249 171 239 182 246 173 240 249 246 i). Compute the least-squares regression line for predicting auditory response time (y) from visual res time (x). ii). Construct a 99% confidence interval for the slope of the least-squares regression line. iii). Test H₀: $\beta _ { 1 }$ =0 versus $\beta _ { 1 } \neq 0$ .Use the $\alpha = 0.01$ =0.01 level of significance.