Sociology 362
Data Exercise 1b
1. For the data on “month,” “leaded gas,” and “cord lead,” the model of interest is
Yi=α+βXi+i(1)
where Y is cord lead, X is leaded gas, and the disturbance ihas mean 0 and variance
σ2(Y|X) = σ2(|x).
a. Write the population regression function that expresses the mean of Y as a function of X.
b. Explain whether the following equation is correct: E(Y|X) = α+β X +.
c. Estimate the conditional variance σ2(Y|X). Is your estimator unbiased? Explain.
d. Find and interpret the “standard error of estimate.” What is the metric (i.e., measurement
units/scale) of the standard error of estimate?
e. Estimate and interpret the standard error of the least-squares regression coefficient ˆ
β. Would
it make sense to estimate the standard error of β?
stop here
The following problems are for later, when we get to statistical inference.
f. Suppose β= 0: what is the probability of observing ˆ
β > 1?
g. Construct the 95% confidence interval for the regression coefficient β.
h. At the .05 level of significance, do a t-test of the null hypothesis β= 0 against the two-sided
alternative. Also test β= 1.6 against the alternative β < 1.6.
i. Carry out the appropriate F-test (α-level .05) of the null model
Yi=α+i(2)
against the alternative
Yi=α+βXi+i(3)
j. What is the relationship between the F-test in “i” and the t-test of β= 0 in problem “g”?
k. Give the point estimate and the 95% confidence interval estimate of the mean µ(Y|X= 140).
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