Homework:
Under the linear model Yi = β0 + β1Xi + ∈i, where ∈i, i = 1, ... , n are independent identical distributed. Assume ∈i ~ N(0, 1). Note that now σ2 = 1 is given.
1, Suppose we would like to know whether α0,β0 + α1β1 = 0
1. write down the hypothesis for testing α0,β0 + α1β1 = 0.
2. construct the test statistics T
3. create a criteria based on T to reject null hypothesis so that the type I error is controlled at α.
4. construct the 1 - α confidence interval
2, We know the s2 = Σi∈^i/n-2, and (n - 2)s2/σ2 ~ xn-22. Suppose we would like to know whether σ2 = 1
1. write down the hypothesis for testing σ2 = 1
2. construct the test statistics T
3. create a criteria based on T to reject null hypothesis so that the type I error is controlled at α.
4. construct the 1 - α confidence interval.
Reading assignment: Read the note on linear algebra.
3.
Let Y1, Y2, Y3 be independent response observations satisfying
μ + β + ∈i i= 1,
Yi =
μ + β + ∈i, i= 2, 3
where μ, β are unknown parameters and ∈1, ∈2, ∈3 are independent N(0, σ2) variables for some unknown σ2 > 0.
(a) Represent the above setting in the form of a simple linear regression model and specify the values of the explanatory variable X for the three observations.
(b) Express the least squares estimates of μ and β in terms of Y1, Y2, Y3.
(c) Express the fitted values of Y1, Y2, Y3 in terms of Y1, Y2, Y3
(d) Express the residual sum of squares SSE in terms of Y1, Y2, Y3. What is the distribution of SSE?
(e) Suppose that (Y1, Y2, Y3) is observed to be (1, -2, 2). (1) Calculate the coefficient of determination R2.
(ii) Conduct an F test to determine if you have evidence in support of the hypothesis that Y1, Y2, Y3 are identically distributed. Give your answer on the basis of a p-value calculated for the F test.
4. Carry out a simple linear regression analysis on the following data.
| Regressor, |
x |
-3 |
-2 |
-1 |
-1 |
0 |
1 |
1 |
2 |
2 |
3 |
| Response, |
Y |
114 |
112 |
110 |
107 |
107 |
105 |
104 |
104 |
101 |
96 |
(a) Find a 90% confidence interval for the true slope of the regression line.
(b) Find a 90% confidence interval for the true y-intercept of the regression line.
(c) Find a 90% confidence interval for σ-, the true standard deviation of Y. [Hint: The residual sum of squares is distributed as cr2x2f for some suitably chosen 1.]
(d) Find a 90% prediction interval for a future observation of Y at x = 1.5.
(e) Find a 90% prediction interval for the average of eight independent future observations of Y at X =1.5
(f) Find a 90% prediction interval for the difference between two future observations of Y, one observed at x = 2.5 and the other at x = 1.5.
(g) Find a 90% prediction interval for a future observation of Y at x = -2000, Comment on the validity of this interval.
5. A random sample of 18 U.S. males was selected, and the following information was recorded for each individual:
x = weight (in g) of fat consumed per day,
y = total cholesterol (in mg) in blood per deciliter.
The data are tabulated as follows:
| Daily fat intake x, (in g) |
29 |
43 |
52 |
56 |
64 |
77 |
81 |
84 |
93 |
| Total cholesterol y, (in nigidl) |
163 |
169 |
136 |
187 |
188 |
176 |
113 |
196 |
240 |
| Daily fat intake x, (in g) |
101 |
105 |
110 |
113 |
120 |
127 |
134 |
148 |
157 |
| Total cholesterol y, (in mg/dl) |
239 |
258 |
283 |
244 |
291 |
298 |
265 |
297 |
320 |
(a) Plot y against x.
(b) Fit a simple linear regression model to the dataset and plot the fitted regression line on the graph obtained in (a).
(c) Compile an ANOVA table for the model fitted in (b). Test at the 5% level whether "daily fat intake" is effective in explaining the variation in cholesterol level among the U.S. males.
(d) Construct a 95% confidence interval for the expected cholesterol level for people whose daily fat intake is 100g.
(e) Construct a 95% prediction interval for the cholesterol level of an individual whose daily fat intake is 100g.
(f) Calculate the coefficient of determination R2 for the simple linear regression model.
(g) A margarine manufacturer claims that the difference between the expected blood choles¬terol level of individuals consuming 100g of fat per day and that of those consuming 40g of fat per day does not exceed 35 mg/dl. If his claim is true, then perhaps some people would be willing to include extra fat in their diets, thinking that the resulting increase in cholesterol is small enough so that there is no need for concern.
Carry out a size 0.05 test for the manufacturer's claim.