National Paper Company must purchase a new machine for producing cardboard boxes. The company must choose between two machines. The machines produce boxes of equal quality, so the company will choose the machine that produces (on average) the most boxes. It is known that there are substantial differences in the abilities of the company's machine operators. Therefore, National Paper has decided to compare the machines using a paired difference experiment. Suppose that eight randomly selected machine operators produce boxes for one hour using machine 1 and for one hour using machine 2, with the following results: Ds BoxYield

	Machine Operator
	1	2	3	4
Machine 1	53	60	58	48
Machine 2	50	55	56	44
	Machine Operator
	5	6	7	8
Machine 1	46	54	62	49
Machine 2	45	50	57	47

Estimate the minimum and maximum differences between the mean outputs of the two machines. Justify your answer.

Asphalt compressive strength is measured in units of psi. To test the repeatability and reproducibility of strength measurements, two engineers in charge of the quality system make two measurements on six specimens. The data are as follows:

$$\begin{matrix} \text{Operator 1} & \quad & \text{Operator 2} & \quad\\ \text{Meas. 1} & \text{Meas. 2} & \text{Meas. 1} & \text{Meas. 2}\\ \text{1501.22} & \text{1510.00} & \text{1505.66} & \text{1504.74}\\ \text{1498.06} & \text{1512.40} & \text{1504.64} & \text{1501.82}\\ \text{1506.44} & \text{1513.54} & \text{1499.84} & \text{1492.95}\\ \text{1496.35} & \text{1541.54} & \text{1502.19} & \text{1507.04}\\ \text{1502.03} & \text{1499.46} & \text{1503.08} & \text{1498.43}\\ \text{1499.90} & \text{1521.83} & \text{1515.57} & \text{1512.84}\\ \end{matrix}$$

(a) Perform an ANOVA on these data. (b) Find $\sigma_{\mathrm{Repeatability}}^{2} \text { and } \sigma_{\mathrm{Reproducibility}}^{2}$ (c) Comment on the significance of the overall gauge capability.

Use the formula $h=\frac{d}{\cot \beta-\cot \alpha}$ to solve the following problem.

A camera operator sits in a stationary crane, 40 feet above ground. He films a scene below of an automobile approaching. At the beginning of the scene, the angle of depression to the driver is $23^{\circ}$. After a direct approach towards the crane, the scene ends with the camera at an angle of $47^{\circ}$. How far did the vehicle travel?

Let $T: V \rightarrow V$ be the linear operator defined by T(v)=Av. a. Show that T is an isomorphism. b. Find $A^{-1}$. c. Show directly that $T^{-1}(\mathbf{w})=A^{-1} \mathbf{w}$ for all $\mathbf{w} \in V$.

$$T\left(\left[\begin{array}{l} x \\ y \end{array}\right]\right)=\left[\begin{array}{rr} -2 & 3 \\ -1 & -1 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]$$

Another application for exponential distributions is supply/demand problems. The operator of a pumping station in a small Wyoming town has observed that demand for water on a typical summer afternoon is exponentially distributed with a mean of 75 cfs (cubic feet per second). Let x be a random variable that represents the town's demand for water (in cfs). What is the probability that on a typical summer afternoon, this town will have a water demand x between 60 and 100 cfs?

Prepare general journal entries to record the transactions below for Spade Company by using the following accounts: Cash; Accounts Receivable; Office Supplies; Office Equipment; Accounts Payable; K. Spade, Capital; K. Spade, Withdrawals; Fees Earned; and Rent Expense. Use the letters beside each transaction to identify entries. After recording the transactions, post them to T-accounts, which serve as the general ledger for this assignment. Determine the ending balance of each T-account.

h. The company collected $1,125 cash as partial payment for the account receivable created in transaction f.

Define a linear operator $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by

$$T\left(\left[\begin{array}{l} x \\ y \end{array}\right]\right)=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]$$

Describe geometrically the action of T on a vector in $\mathbb{R}^{2}$.

(Program) a. For the Time class constructed in Exercise 10 of Section 11.2, remove the display function and include overloaded extraction and insertion operator functions for the input and output of Time objects, using cin and cout. Times should be displayed in the form hrs:min:sec. b. Include the overloaded functions written for Exercise 8a in a Time class, and then write, compile, and run a C++ program to verify that the overloaded functions input and output a Time object correctly.

Rewrite the following function to use pointer arithmetic instead of array subscripting. (ln other words, eliminate the variables i and j and all uses of the [ ] operator.) Use a single loop instead or nested loops.

int sum_two_dimensional_array(const int a[][LEN], int n) 
{
int i, j, sum - 0;
for (i = 0; i < n; i ++)
  for(j = 0; j < LEN; j ++)
    sum + = a[i][j];
return sum;
}

Lou Goble, an internal auditor for a large manufacturing enterprise, received an anonymous note from an assembly-line operator who has worked at the company’s West Coast factory for the past 15 years. The note indicated that there are some fictitious employees on the payroll as well as some employees who have left the company. He offers no proof or names. What CAAT could Lou use to substantiate or refute the employee’s claims? (CIA Examination, adapted)

Prepare general journal entries to record the transactions below for Spade Company by using the following accounts: Cash; Accounts Receivable; Office Supplies; Office Equipment; Accounts Payable; K. Spade, Capital; K. Spade, Withdrawals; Fees Earned; and Rent Expense. Use the letters beside each transaction to identify entries. After recording the transactions, post them to T-accounts, which serve as the general ledger for this assignment. Determine the ending balance of each T-account.

d. The company received $15,500 cash as fees for services provided to a customer.

Marsala, a college student, signs a oneyear lease agreement that runs from September 1 to August 31. The lease agreement specifies that the lease cannot be assigned without the landlord’s consent. In late May, Marsala decides not to go to summer school and assigns the balance of the lease (three months) to a close friend, Fred. The landlord objects to the assignment and denies Fred access to the apartment. Marsala claims that Fred is financially sound and should be allowed the full rights and privileges of an assignee. Discuss fully who is correct, the landlord or Marsala. (See Assignments and Delegations.)

Consider a physical system whose Hamiltonian $H$ and an operator $A$ are given, in a threedimensional space, by the matrices

$$\begin{align*} H=\hbar \omega\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array}\right), \quad A=a\left(\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right) . \end{align*}$$

Let $g(t)$ be the auxiliary polynomial associated with a homogeneous linear differential equation with constant coefficients and let $V$ denote the solution space of this differential equation. Prove the following results. $\\$ (a)$$\mathrm{V} is a D-invariant subspace, where$$\mathrm{D} is the differentiation operator on$$\mathrm{C}$^${\infty} .$ $(b) The minimal polynomial of D_{(the restriction of } D to V ) is g ( t ) .$ $(c) If the degree of$g(t)$is$n,$then the characteristic polynomial of$$\mathrm{D}$_${\mathrm{V}} is$(-1)^{n} g(t) .$