An auto insurance company has 10,000 policyholders. Each policyholder is classified as: (i) young or old; (ii) male or female; and (iii) married or single. Of these policyholders, 3000 are young, 4600 are male, and 7000 are married. The policyholders can also be classified as 1320 young males, 3010 married males, and 1400 young married persons. Finally, 600 of the policyholders are young married males. How many of the company's policyholders are young, female, and single? Choose one of the following. (a) 280 (b) 423 (c) 486 (d) 880 (e) 896

Viewing Habits A survey of a group's viewing habits over the last year revealed the following information:
(i) $28 \%$ watched gymnastics;
(ii) $29 \%$ watched baseball;
(iii) $19 \%$ watched soccer;
(iv) $14 \%$ watched gymnastics and baseball;
(v) $12 \%$ watched baseball and soccer;
(vi) 10% watched gymnastics and soccer;
(vii) $8 \%$ watched all three sports.
Calculate the percentage of the group that watched none of the three sports during the last year. Choose one of the following. Source: Society of Actuaries.
a. 24
b. 36
c. 41
d. 52
e. 60

A machine that fills quart orange juice cartons is set to fill them with 32.1 oz. If the actual contents of the cartons vary normally, with a standard deviation of 0.1 oz, what percentage of the cartons contain less than a quart (32 oz)?

Use the form $y_{p}=a \cos \omega t+b \sin \omega t$, to help find a particular solution for the given differential equation. $y^{\prime \prime}+7 y^{\prime}+6 y=3 \sin 2 t$

An insurance company has written 100 policies for $100,000,500 policies for$50,000 and 1000 policies for $10,000 for people of age 20. If experience shows that the probability that a person will die at age 20 is 0.0012, how much can the company expect to pay out during the year the policies were written?

Let f(x)=x-1, $g(x)=\sqrt{x+1}$, and $h(x)=2 x^{3}-1$. Find the rule for each function. $\frac{g h}{g-f}$

Construct a truth table for each statement. Is the statement a tautology? $\sim(p \wedge q) \rightarrow(\sim p \vee \sim q)$

The article “Optimization of Surface Roughness in Drilling Using Vegetable-Based Cutting Oils Developed from Sunflower Oil” (Industrial Lubrication and Tribology, 2011: 271–276) gave the following data on $x_1$ = spindle speed (rpm), $x_2$ = feed rate (mm/rev), $x_3$ = drilling depth (mm), and y = surface roughness $(\mu \mathrm{m})$ when a semisynthetic cutting fluid was used:

$$\begin{matrix} \ {x_1} & \ {x_2} & \ {x_3} & \text{y} & \ {e^{*}}\\ \text{320} & \text{.10} & \text{15} & \text{2.27} & \text{-1.32}\\ \text{320} & \text{.12} & \text{20} & \text{4.14} & \text{1.08}\\ \text{320} & \text{.14} & \text{25} & \text{4.69} & \text{0.26}\\ \text{420} & \text{.10} & \text{20} & \text{1.92} & \text{-0.40}\\ \text{420} & \text{.12} & \text{25} & \text{2.63} & \text{-0.79}\\ \text{420} & \text{.14} & \text{15} & \text{4.34} & \text{0.99}\\ \text{520} & \text{.10} & \text{25} & \text{2.03} & \text{1.64}\\ \text{520} & \text{.12} & \text{15} & \text{2.34} & \text{0.03}\\ \text{520} & \text{.14} & \text{20} & \text{2.67} & \text{-1.52}\\ \end{matrix}$$

The estimated standard deviation of the predicted Y when $x_1 = 400$ and $x_2 = .125$ is .180. Calculate a 95% CI for true average roughness under these circumstances.

Rationalize the denominator: $\frac{\sqrt{x}-1}{2 \sqrt{x}}$.

Find the coordinate vector $\mathbf{x}_{B}$ corresponding to x and basis $\mathcal{B}$.

$$\mathcal{B}=\left\{\left[\begin{array}{l} 1 \\ 2 \end{array}\right],\left[\begin{array}{l} 2 \\ 5 \end{array}\right]\right\}, \quad \mathbf{x}=\left[\begin{array}{r} 3 \\ -2 \end{array}\right]$$

Solve the equation and check your solution. $\sqrt{t}=4$

An article in Lubrication Engr. (“Accelerated Testing of Solid Film Lubricants,” 1972: 365–372) reported on an investigation of wear life (y, in hr) for solid film lubricant. Three sets of journal bearing tests were run on a Mil-L-8937-type film at each combination of three speeds ($x_1$, in rpm), and three loads ($x_2$, in 1000s of hr). The values of $x_1$ for the resulting 27 observations were 20, 20, ... , 20, 60, ... , 60, 100, ... , 100, and the values of $x_2$ were 3, 3, 3, 6, 6, 6, 10, 10, 10, 3, 3, 3, ... , 10, 10, 10. The corresponding values of y = wear life (hr) were 300.2, 310.8, 333.0, 99.6, 136.2, 142.4, 20.2, 28.2, 102.7, 67.3, 77.9, 93.9, 43.0, 44.5, 65.9, 10.7, 34.1, 39.1, 26.5, 22.3, 34.8, 32.8, 25.6, 32.7, 2.3, 4.4, and 5.8. The investigators commented that a lognormal distribution is appropriate for Y because ln(Y) is known to follow a normal law, and then proposed the multiplicative power regression model $Y=\alpha x_{1}^{\beta_{1}} x_{2}^{\beta_{2}} \epsilon$. Does it appear that both predictors provide useful information about wear life?

A company is taking a survey to find out whether people like its product. Its last survey indicated that 70% of the population like the product. Based on that, in a sample of 58 people, find the probabilities of the following. (a) All 58 like the product. (b) From 28 to 30 (inclusive) like the product.

Solve the equation and check your solution. $\sqrt{x}-10=0$