1. Mahrita.Harahap@uts.edu.au Maths Study Centre CB04.03.331 Open 11am – 5pm Semester Weekdays https://www.uts.edu.au/future-students/science/student-experience/maths-study- centre https://www.uts.edu.au/future-students/science/student-experience/maths-study- centre www.khanacademy.org www.mahritaharahap.wordpress.com/teachingareas/regression START>ALL PROGRAMS>IBM SPSS>IBM SPSS STATISTICS 19 Marking Scheme: 0 if less than 50% attempted, 1 for more than 50% attempted but less than 50% correct, 2 if more than 50% correct.

3. The assumptions of the model is that the random errors (residuals) are normally distributed and random (have constant variance). 1.Normality Assumption: look at the histogram or Normal P-P plot. The normal probability plot is constructed by plotting the expected values of the residuals under the normality assumption (line) against the actual values of the residuals. If the normal assumption is valid, the residuals should lie approximately on the straight line. Any non-linear trend indicates the violation of the normal assumption. 2.Constant Variance Assumption: look at the residuals vs fits plot (GRAPH>LEGACYDIALOG>SCATTER>SIMPLESCATTER>Y AXIS:STUDENTISED RESIDUAL>X AXIS: PREDICTED VARIABLE. Residuals should have a constant variance (does not exhibit a pattern/random scatter). If the variance is not constant (patterns or increasing variance) then ordinary least squares is not the most efficient estimation method to characterise the relationship.

4. d) Use t-test when testing individual parameters: For obs 1: H 0 : α=0. The intercept is 0. H 1 : α ≠0. The intercept is not equal to 0. Test Statistic: Since p-value=0.003<0.05 we reject H o. We have enough evidence to prove it takes α ≠0 at the 5% level of significance. H 0 : β=1. The population slope is 1. H 1 : β≠1. The population slope is not equal to 1. Test Statistic: If |t|>t n-2, p-value≤0.05. Reject H o. We have enough evidence to prove β≠1 at the 5% level of significance. If |t| 0.05. Do not reject H o. We do not have enough evidence to prove β≠1 at the 5% level of significance. Since |t|<t n-2 we do not reject H o and conclude we do not have enough evidence to prove β≠1 at the 5% level of significance. For obs 2: H 0 : α=0. The intercept is 0. H 1 : α ≠0. The intercept is not equal to 0. Since p-value=0.003<0.05 we reject H o. We have enough evidence to prove it takes α ≠1 at the 5% level of significance. H 0 : β=1. The population slope is 1. H 1 : β≠1. The population slope is not equal to 1. Test Statistic: If |t|>t n-2, p-value≤0.05. Reject H o. We have enough evidence to prove β≠1 at the 5% level of significance. So Observer 2 seems more biased in their estimates given a slope significantly different from 1, even though the R 2 for observer 2 is greater than observer 1.

6. If we look at the scatterplot of the residuals vs. fitted graph and see that the variation increases – we have non-constant variance (constant variance assumption is not valid. If the variance is not constant (exhibits a pattern) then ordinary least squares is not the most efficient estimation method We turn to WEIGHTED REGRESSION!

7. For obs 1: H 0 : α=0. The intercept is 0. H 1 : α ≠0. The intercept is not equal to 0. Test Statistic: Since p-value=0.038<0.05 we reject H o. We have enough evidence to prove it takes α ≠0 at the 5% level of significance. H 0 : β=1. The population slope is 1. H 1 : β≠1. The population slope is not equal to 1. Test Statistic: If |t|>t n-2, p-value≤0.05. Reject H o. We have enough evidence to prove β≠1 at the 5% level of significance. If |t| 0.05. Do not reject H o. We do not have enough evidence to prove β≠1 at the 5% level of significance. Since |t|<t n-2 we do not reject H o and conclude we do not have enough evidence to prove β≠1 at the 5% level of significance.

15. e) Forward Selection Models: Backward Selection Models: