An actuary studying the insurance preferences of automobile owners makes the following conclusions: (i) An automobile owner is twice as likely to purchase collision coverage as disability coverage. (ii) The event that an automobile owner purchases collision coverage is independent of the event that he or she purchases disability coverage. (iii) The probability that an automobile owner purchases both collision and disability coverages is 0.15. What is the probability that an automobile owner purchases neither collision nor disability coverage? Choose one of the following. (a) 0.18 (b) 0.33 (c) 0.48 (d) 0.67 (e) 0.82

Data from the National Longitudinal Survey of Youth (NLSY) were analyzed to derive the probabilities that children will repeat the child-raising choices of their parents. The living arrangements of a group of children at age 14 were recorded. The NLSY then tracked those same children and recorded the living arrangements for their children. Based on these data, the following transition matrix gives the probabilities of change in child-raising choices from one generation to the next. Find the long-term proportions of child-raising choices if this trend were to continue.

$$\underline{\text{Second-Generation Living Arrangements}}\\ \begin{array}{lccccc} & \text { Couple } & \text { Mother } & \text { Father } & \text { Relative } & \text { Other } \\ \hline \text {Couple} & 0.566 & 0.309 & 0.083 & 0.023 & 0.019 \\ \text {Mother} & 0.392 & 0.453 & 0.066 & 0.061 & 0.028 \\ \text {Father} & 0.391 & 0.354 & 0.109 & 0.108 & 0.038 \\ \text {Relative} & 0.307 & 0.558 & 0.040 & 0.056 & 0.039 \\ \text {Other} & 0.337 & 0.320 & 0.025 & 0.252 & 0.066 \end{array}$$

An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is $0.44$. Calculate the number of blue balls in the second urn. Choose one of the following. Source: Society of Actuaries.
a. 4
b. 20
c. 24
d. 44
e. 64

An urn contains 10 balls: 4 red and 6 blue. A second urn contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is 0.44. Calculate the number of blue balls in the second urn. Choose one of the following. (a) 4 (b) 20 (c) 24 (d) 44 (e) 64

Solve each equation by factoring or by using the quadratic formula. If the solutions involve square roots, give both the exact solutions and the approximate solutions to three decimal places. $3 k^{2}+k=6$

A medical laboratory tested 21 samples of human blood for acidity on the pH scale, with the following results.

$$\begin{array}{lllllll} 7.1 & 7.5 & 7.3 & 7.4 & 7.6 & 7.2 & 7.3 \\ 7.4 & 7.5 & 7.3 & 7.2 & 7.4 & 7.3 & 7.5 \\ 7.5 & 7.4 & 7.4 & 7.1 & 7.3 & 7.4 & 7.4 \end{array}$$

(a) Find the mean and standard deviation. (b) What percentage of the data is within 2 standard deviations of the mean?

Suppose the true average growth m of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is $\sigma^{2}$, whereas for the second type the variance is $4\sigma^{2}$. Let $X_1, ... , X_m$ be m independent growth observations on the first type [so $E\left(X_{i}\right)=\mu, V\left(X_{i}\right)=\sigma^{2}$], and let $Y_1, ... , Y_n$ be n independent growth observations on the second type [$E\left(Y_{i}\right)=\mu, V\left(Y_{i}\right)=4 \sigma^{2}$]. For fixed m and n, compute $V(\hat{\mu})$, and then find the value of $\delta$ that minimizes $V(\hat{\mu})$. [Hint: Differentiate $V(\hat{\mu})$ with respect to $\delta$.]

Simplify the radical expression.

$$\sqrt[3]{32 x^{2} y^{5}}$$

Solve each system of equations by using the inverse of the coefficient matrix if it exists and by the Gauss-Jordan method if the inverse doesn't exist.

$$\begin{aligned} 2 x+y &=1 \\ 3 y+z &=8 \\ 4 x-y-3 z &=8 \end{aligned}$$

Use the Ad ams-Bashforth-Moulton method to approximate y( 1.0 ), where y (x) is the solution of the given initial-value problem. First use h = 0.2 and then use h = 0.1 . Use the RK4 method to compute $y_{1}, y_{2}, \text { and } y_{3}$. $y^{\prime}=1+y^{2}, \quad y(0)=0$

A city has 8,958 recycle bins. The city gives half of the recycle bins to its citizens. The rest of the recycle bins are divided into 25 equal groups for city parks. How many recycle bins are left over?

Construct a truth table for each compound statement. $\sim q \wedge(\sim p \vee q)$

Simplify the expression.

$$\frac{5}{9-\sqrt{6}}$$

An experiment was carried out to investigate the effect of species (factor A, with I = 4) and grade (factor B, with J = 3) on breaking strength of wood specimens. One observation was made for each species—grade combination— resulting in SSA = 442.0, SSB = 428.6, and SSE = 123.4. Assume that an additive model is appropriate. Test $H_{0}: \beta_{1}=\beta_{2}=\beta_{3}=0$ (no differences in true average strength due to grade) versus $H_a:$ at least one $\beta_{j} \neq 0$ using a level .05 test.