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# 数学代写｜MATH 484/564 HOMEWORK #4

Problem 1 Exercise 4.8 (b) from the TEXT.

Problem 2 Given data in Table 6.2 (attached), the response variable is nt, representing
the number of surviving bacteria (in hundreds) after being exposed to X-ray for t intervals.
The predictor variable is t.

1) First regress nt on time t, plot residuals against the fitted values n^t. Conclude if
the relationship between the mean response and the predictor is linear.

2) Use data transformation on the response variable, i.e., regress log(nt) on t.
• What is the regression line equation?
• Plot residuals against the fitted values, and conclude if the violation of the ”L”
assumption still exists.

Problem 3 Given data in Table 6.6 (attached), the response variable Y is the number
of injury incidents, and the predictor variable N is the proportion of flights.

1) First regress Y on N, plot residuals against the fitted values Y^ . Conclude if error
is heteroscedastic, i.e., the ”E” assumption is violated.

2) Use data transformation on the response variable, i.e., regress pY on N. The
rationale behind this transformation is that the occurrence of accidents, Y , tends to
follow the Poisson probability distribution, and the variance of pY is approximately
equal to 0.25, see Table 6.5.

• What is the regression line equation?

• Plot residuals against the fitted values, and conclude if there is still evidence
of heteroscedasticity.

Problem 4 Given the data in Table 6.9 (attached), the response variable Y is the
number of supervisors, and the predictor variable X is the number of supervised workers.
Based on empirical observation, it is hypothesized that the standard deviation of the error
term i is proportional to xi: • Use the weighted least squares (WLS) method to fit the model. Provide the regression
equation.

• Use data transformation method to transform Y to Y 0 = Y=X, and transform X to
X0 = 1=X (see equations 6.11 and 6.12), and then use the ordinary least squares
(OLS) method to regress Y 0 on X0. Provide the regression equation.

• Compare the results from the above two methods and conclude if the two methods
are equivalent. You can compare the residual vs fitted value plot side by side and
conclude if they have the same effect in terms of removing heteroscedasticity.

Problem 5 For the data in Problem 4, use OLS without data transformation to fit
the model, i.e., directly regress Y on X, and compare the variances of the coefficients
Var(β^0) and Var(β^1) with their counterparts obtained by using WLS, conclude which
method yields smaller variances.