91Ó°ÊÓ

A simple curve that often makes a good model for the variable costs of a company, as a function of the sales level \(x\), has the form \(y=\beta_{1} x+\beta_{2} x^{2}+\beta_{3} x^{3} .\) There is no constant term because fixed costs are not included. a. Give the design matrix and the parameter vector for the linear model that leads to a least-squares fit of the equation above, with data \(\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)\) b. [M] Find the least-squares curve of the form above to fit the data \((4,1.58),(6,2.08),(8,2.5),(10,2.8),(12,3.1)\) \((14,3.4),(16,3.8),\) and \((18,4.32),\) with values in thou- sands. If possible, produce a graph that shows the data points and the graph of the cubic approximation.

In Exercises \(9-12,\) find a unit vector in the direction of the given vector. $$ \left[\begin{array}{r}{-30} \\ {40}\end{array}\right] $$

Find an orthogonal basis for the column space of each matrix in Exercises \(9-12\) . $$ \left[\begin{array}{rrr}{1} & {2} & {5} \\ {-1} & {1} & {-4} \\ {-1} & {4} & {-3} \\ {1} & {-4} & {7} \\ {1} & {2} & {1}\end{array}\right] $$

In Exercises \(9-12,\) find a unit vector in the direction of the given vector. $$ \left[\begin{array}{c}{8 / 3} \\ {2}\end{array}\right] $$

Involve a design matrix \(X\) with two or more columns and a least-squares solution \(\hat{\beta}\) of \(\mathbf{y}=X \beta .\) Consider the following numbers. (i) \(\|X \hat{\boldsymbol{\beta}}\|^{2}-\) the sum of the squares of the "regression term." Denote this number by \(\mathrm{SS}(\mathrm{R})\). (ii) \(\|\mathbf{y}-X \hat{\boldsymbol{\beta}}\|^{2}-\) the sum of the squares for error term. Denote this number by \(\mathrm{SS}(\mathrm{E})\). (iii) \(\|\mathbf{y}\|^{2}-\) the "total" sum of the squares of the \(y\) -values. Denote this number by \(\mathrm{SS}(\mathrm{T}) .\) Every statistics text that discusses regression and the linear model \(\mathbf{y}=X \boldsymbol{\beta}+\boldsymbol{\epsilon}\) introduces these numbers, though terminology and notation vary somewhat. To simplify matters, assume that the mean of the \(y\) -values is zero. In this case, SS(T) is proportional to what is called the variance of the set of \(y\) -values. Justify the equation \(\mathrm{SS}(\mathrm{T})=\mathrm{SS}(\mathrm{R})+\mathrm{SS}(\mathrm{E})\). [Hint: Use a theorem, and explain why the hypotheses of the theorem are satisfied.] This equation is extremely important in statistics, both in regression theory and in the analysis of variance.

Let \(\mathbf{u}=\left[\begin{array}{r}{5} \\ {-6} \\ {7}\end{array}\right],\) and let \(W\) be the set of all \(\mathbf{x}\) in \(\mathbb{R}^{3}\) such that \(\mathbf{u} \cdot \mathbf{x}=0 .\) What theorem in Chapter 4 can be used to show that \(W\) is a subspace of \(\mathbb{R}^{3} ?\) Describe \(W\) in geometric language.

Let \(W\) be a subspace of \(\mathbb{R}^{n},\) and let \(W^{\perp}\) be the set of all vectors orthogonal to \(W .\) Show that \(W^{\perp}\) is a subspace of \(\mathbb{R}^{n}\) using the following steps. a. Take \(\mathbf{z}\) in \(W^{\perp}\) , and let u represent any element of \(W .\) Then \(\mathbf{z} \cdot \mathbf{u}=0 .\) Take any scalar \(c\) and show that \(c \mathbf{z}\) is orthogonal to \(\mathbf{u} .\) (Since \(\mathbf{u}\) was an arbitrary element of \(W,\) this will show that \(c \mathbf{z}\) is in \(W^{\perp} . )\) b. Take \(\mathbf{z}_{1}\) and \(\mathbf{z}_{2}\) in \(W^{\perp},\) and let \(\mathbf{u}\) be any element of \(W .\) Show that \(\mathbf{z}_{1}+\mathbf{z}_{2}\) is orthogonal to \(\mathbf{u} .\) What can you conclude about \(\mathbf{z}_{1}+\mathbf{z}_{2} ?\) Why? c. Finish the proof that \(W^{\perp}\) is a subspace of \(\mathbb{R}^{n}\) .