Example Model Utility Test 8 Pvalue The table

Example - Model Utility Test 8. P-value: The table of tail areas for tdistributions only has t values 4, so we can see that the corresponding tail area is < 0. 002. Since this is a two-tail test the P-value < 0. 004. (Actual calculation gives a P-value = 0. 002) 2

Example - Model Utility Test 8. Conclusion: Even though no specific significance level was chosen for the test, with the P-value being so small (< 0. 004) one would generally reject the null hypothesis that b = 0 and conclude that there is a useful linear relationship between the % unemployed and the suicide rate. 3

Example - Minitab Output 4

Residual Analysis The simple linear regression model equation is y = a + bx + e where e represents the random deviation of an observed y value from the population regression line a + bx. Key assumptions about e 1. At any particular x value, the distribution of e is a normal distribution 2. At any particular x value, the standard deviation of e is s, which is constant over all values of x. 5

Residual Analysis To check on these assumptions, one would examine the deviations e 1, e 2, …, en. Generally, the deviations are not known, so we check on the assumptions by looking at the residuals which are the deviations from the estimated line, a + bx. The residuals are given by 6

Standardized Residuals Recall: A quantity is standardized by subtracting its mean value and then dividing by its true (or estimated) standard deviation. For the residuals, the true mean is zero (0) if the assumptions are true. 7

Standardized Residuals As you can see from the formula for the estimated standard deviation the calculation of the standardized residuals is a bit of a calculational nightmare. Fortunately, most statistical software packages are set up to perform these calculations and do so quite proficiently. 8

Standardized Residuals - Example Consider the data on percentage unemployment and suicide rates Notice that the standardized residual for Pittsburgh is -2. 50, somewhat large for this size data set. 9

Example Pittsburgh This point has an unusually high residual 10