7. Residual analysis Aa Aa Consider a multiple regression model of the dependent

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7. Residual analysis Aa Aa Consider a multiple regression model of the dependent variable y on independent variables xl, X2, and X3: Using data with n 12 observations for each of the variables, a researcher obtains the following estimated regression equation for the above model 0.5216 1.2419x1 + 0.3049x2-0.0217x3 The standard error of estimate for this equation is s0.6489 The table below gives the values for the independent and dependent variables and their corresponding predicted values, residuals, and leverage Predicted Value Residua Leverage No X3 (y - y) 0.9326 1.4814 -0.3408 7.4175 0.3461 0.1632 0.0768 13.5768-1.2659 0.0536 0.4631 0.2575 12.7237 -0.1821 7.5493 -0.2890 0.4771 0.4520 0.1435 0.0988 0.2440 0.0738 0.2514 0.4710 0.0440 0.3518 0.3679 0.0936 0.4082 15.0042 7.9611 15.0243 -0.6987 15.9368 1.0986 -1.3892-0.8676 1.1405 3 -5.4086-3.9838 0.3419-7.7636 3.16001.1120 1.1186 4.3239 0.3301 0.2630 7.685311.5261 0.1471 12.3110 7-1.7893 10.0129 1.2846 1.3775 3.5670 0.3263 1.3994 7.6694-1.8849 -0.2815 9.7355 4.89370.3933 12.5416 11 -4.8492-6.7252-0.0790-7.8382 12 -1.0189-10.2205 0.7441 -3.3985 4 -0.4070 51.3257 6 0.9553 0.1862 1.3240 0.9364 9.4779 8 -0.5359 10 8.6307 -3.8757 Use information from the table to answer the following questions

Explanation / Answer

Standard error : s = 0.6489
Standard deviation ( residual ) : = (s2 * 8/11) = 0.55338
Leverage for the 10th observation : h10 = 0.3679
Residual for the 10th observation : e10 = -0.1821
The standardized residual for the tenth observation : r10 = e10/{ (1-h10)} = -0.413898.

Revised standard error : s(10) = 0.2122
Revised standard deviation of the residual is : (10) = 0.18096. [ by the same formula]
Studentized deleted residual for the tenth observation : r10* = e10/{ (10) (1-h10)} = -1.26571.

Under the null hypothesis that the tenth observation is not an outlier, it’s studentized deleted residual follows t distribution with (12-3-1) = 8 degrees of freedom.
At significance level =0.05, the critical values for the test of the null hypothesis using the studentized deleted residual are 2.306004, - 2.306004. [ Critical values = upper and lower 100% points of t-distribution with df 8.]
Thus, you do not conclude that the tenth observation is an outlier.

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