Math Problems Vault

Problem 7.1

Using the method of orthogonal polynomials described in Section 7.1.2, fit a third-degree equation to the following data:

\begin{array}{llllll} y \text{ (index): } & 9.8 & 11.0 & 13.2 & 15.1 & 16.0 \\ x \text{ (year): } & 1950 & 1951 & 1952 & 1953 & 1954 \end{array}

Test the hypothesis that a second-degree equation is adequate.

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Problem 7.2

Show that the least squares estimates of $\beta_{1}$ and $\beta_{2}$ for the model (7.28) are still unbiased even when the true model includes an interaction term $\beta_{12}$ , that is,

\mathbb {E}\left[Y\right] = \beta _{0}+\beta _{1} x_{1}+\beta _{2} x_{2}+\beta _{12} x_{1} x_{2}

Find the least squares estimate of $\beta_{12}$ .

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Problem 7.3

Suppose that the regression curve

\mathbb {E}\left[Y\right] = \beta _{0}+\beta _{1} x+\beta _{2} x^{2}

has a local maximum at $x = x_{m}$ where $x_{m}$ is near the origin. If $Y$ is observed at $n$ points $x_{i}(i = 1,2, \ldots , n)$ in $\left[-a, a\right]$ , $\bar{x} = 0$ , and the usual normality assumptions hold, outline a method for finding a confidence interval for $x_{m}$ . Hint: Use the method of Section 6.1.2.

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