Homework2Key

5) Compute by hand a 95% confidence interval for mean pain when anxiety = 4 and a 95% prediction interval (PI) for pain when anxiety = 4. Briefly contrast the CI vs. PI. 95% Confidence Interval (you might get slightly different answer depending on what value you use for the t statistic) ¿ ^ μ ( x 0 ) ±t n − 2,0.975 ^ σ √ 1 n + ( x 0 − x ) 2 SSX ¿ ( 2.696 + 0.485 ∗ 4 ) ± 1.972 ∗ 2.173 √ 1 199 + ( 4 − 2.111 ) 2 1390 ¿ ( 4.26 , 5.01 ) 95% Prediction Interval (you might get slightly different answer depending on what value you use for the t statistic) ¿ ^ μ ( x 0 ) ±t n − 2,0.975 ^ σ √ 1 + 1 n + ( x 0 − x ) 2 SSX ¿ ( 2.696 + 0.485 ∗ 4 ) ± 1.972 ∗ 2.173 √ 1 + 1 199 + ( 4 − 2.111 ) 2 1390 ¿ ( 0.33,8.94 ) Prediction intervals must account for both the uncertainty in knowing the value of the population mean as well as the scatter of the data (the additional ‘1’ under the radical). That’s why the prediction interval is wider than the confidence interval. 6) What fraction of the variation in pain is explained by anxiety ? 𝑅 2 = 0.2604 26.04% of the variation in pain is explained by anxiety. 7) What fraction of the variation in anxiety is explained by the variation in pain ? Compare your answer to this problem with what you found in the problem above. 3