PROBLEM 1. The following data were collected for a class of 6 students on reading achievement stanines (Y) at the end of first grade and reading readiness stanines (X) at the end of kindergarten. You want to examine how well the readiness scores predict reading achievement.
Y: 3 2 1 3 5 4 X: 2 2 1 1 3 3
DO THE FOLLOWING BY HAND!!!
1. Calculate the least square estimates of the intercept and slope of the regression line of Y on X.
2. Specify the regression equation using your calculated estimates.
4. Calculate the residuals (ei) corresponding to the predicted and observed values of Y. Make a column of these values and find their sum.
5. Calculate the Mean Squared Error (MSE). Do not round your answer.
6. Test the hypotheses H0: Bo = 0 and H0: B1 = 0 at α = 0.05. Find the range of the p-value for both tests.
7. Compute the 95% confidence interval for the slope and give interpretation.
3. Calculate the predicted values of Y associated with the sample values of X (i.e., calculate Yi). Make a column of these values and find their sum.
8. Compute the 95% confidence interval E(YX=3) and give interpretation of the interval.
9. Compute the 95% prediction interval for YX =3 and give interpretation of the interval.
10. Using your estimate of the slope, Sxxand the residuals, calculate SSR, SSE and SST and calculate the coefficient of determination. Provide the appropriate interpretation of this coefficient of determination as it relates to this problem.
DO THE FOLLOWING BY COMPUTER!!!
11. Using Microsoft Excel find the least squares estimate of the regression line for the following set of ten (10) x, y data points. Include the output as identified below.
Y: 2 2 1 1 3 4 5 5 7 6
X: 3 1 1 3 5 4 7 6 7 8
a. The regression estimates including coefficients, SE, t-stats and p-values.
b. ANOVA output.
c. The 90% confidence intervals for X = 3.
MATRIX METHODS
12. Using the data for Problem 1, write in proper matrix form, the matrices for Y, and X.
13. Using matrix methods show your work in finding the estimates for B.
14. From your estimates in 13 write the regression equation.