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- The weight (X) and systolic blood pressure (Y ) of 26 randomly selected males in the age group 25-30 are shown below. Assume that weight and blood pressure are jointly normally distributed.
Weight 165 167 180 155 212 175 190 210 200 Systolic Blood Pressure 130 133 150 128 151 146 150 140 148 Weight 149 158 169 170 172 159 168 174 183 Systolic Blood Pressure 125 133 135 150 153 128 132 149 158 Weight 215 195 180 143 240 235 192 187 Systolic Blood Pressure 150 163 156 124 170 165 160 159
(a) Find a regression line regarding systolic blood pressure to weight.
(b) Set up the ANOVA table. What proportion of the variation in Y is accounted for by introducing X into the regression model?
(c) Determine r, r^2 and interpret your results.
(d) Test the hypothesis that H 0 : ρ = 0 versus H 1 : ρ > 0 at α = 0.05.
(e) Test the hypothesis that H 0 : ρ = 0.6 versus H 1 : ρ 6 = 0.6 at α = 0.05.
(f ) Find a 95% confidence interval for ρ.
(g) Fit a no-intercept model to the data and compare it to the model obtained in part (a). Which model would you conclude is superior?
- Complete the following ANOVA table. What does the p-value imply?
Source df Sum of squares Mean of squares F -value p-value Model 1
Error 0.
Total 19 13.
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- The purity of oxygen produced by a fractionation process is thought to be related to the percentage of hydrocarbons in the main condenser of the processing unit. Twenty samples are shown below.
Purity (%) 86.91 89.85 90.28 86.34 92.58 87.33 86. Hydrocarbon (%) 1.02 1.11 1.43 1.11 1.01 0.95 1. Purity (%) 91.86 95.61 89.86 96.73 99.42 98.66 96. Hydrocarbon (%) 91.86 95.61 89.86 1.46 1.55 1.55 1. Purity (%) 93.65 87.31 95.00 96.85 85.20 90. Hydrocarbon (%) 1.40 1.15 1.01 0.99 0.95 0.
(a) Fit a simple linear regression model to the data when purity of oxygen regressed on hydrocarbon.
(b) Set up the ANOVA table. Use the F -test for the significance of the straight- line regression.
(c) Test the hypothesis H 0 : β 1 = 0 versus H 1 : β 1 6 = 0 at α = 0.05, by using a t-test.
(d) Compare the value of the test statistics F obtained in part (b) with the value of T 2 , the square of the test statistic obtained in part (c). Are the p-value for the F -test in part (b) and for the t-test in part (c) the same? If yes, why does this make sense intuitively?
- Consider the simple linear regression model
Y = β 0 + β 1 X + E
with E(E) = 0, Var(E) = σ^2 , and E uncorrelated. Show that
E (M SR) = σ^2 + β^21 Sxx and E (M SRes) = σ^2.
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