[SOLVED] STAT3600 Linear Statistical Analysis 24 Statistics

STAT3600 Linear Statistical Analysis

1.   [49] Consider the data of five observations.

a.   [5]  Write  down the simple linear regression model of yi  on xi .  What  are the four model assumptions? State them clearly.

b.   [5] Letβ(^)1  be the least squares estimator for the unknown population slope in the simple linear

regression model. Prove that 

c.   [5] Find the least squares estimates of the population intercept and slope. Interpret the estimate for the population slope.

d.   [15] Construct the following ANOVA table by filling in the blanks led by letters from A to I.

At 5% significance level, test whether there is a linear relationship between the independent and dependent variables using the information on the ANOVA table. State clearly the null and alternative hypotheses, test statistic, null distribution, decision rule and conclusion.

e.   [6] Using the Bonferroni’s method, construct simultaneous confidence intervals for the

population intercept and slope with a family confidence level of at least 95%.

f.    [2] Find the coefficient of determination and interpret the result.

g.   [1] Find a point estimate for the population mean of Y when x is 25.

h.   [4] Construct a 90% confidence interval for the population mean of Y when x is 25.

i.    [6] Let Y(1)  and Y(2)  be future responses with the values of x being 30 and 35, respectively. Construct a 95% prediction interval for Y(1)  − Y(2) .

2.   [51] You are given the following matrices computed from a multiple linear regression of yi  = β0 + β1xi1 + β2xi2 + εi:

The matrices are properly ordered according to the regression equation given above.

a.   [4] Find the sample size and the sample mean of r.

b.   [5] Show that the least squares estimator for β is given by β(^) = (XTX)-1XTY.

c.   [5] Find the least squares estimates for β0, β1 and β2. Interpret the estimates for β1 and β2.

d.   [15] Construct the ANOVA table and hence, test whether the coefficients for the independent variables are jointly equal to zero at the 5% level of significance. Clearly define the null and alternative hypotheses and decision rule. State your conclusion.

e.   [7] At the 5% level of significance, conduct a t-test for H0 : β1  = β2  vs. H1 : β1  ≠ β2.

f.    [6] Construct a 95% confidence interval for β1  + 2β2.

g.    [9] Define    At  the  5%  level  of  significance,  test  the following hypotheses.

H0 : Cβ = d vs. H1 : Cβ ≠ d.