Consider the $2^{6-2}$ design in table below. Suppose that after analyzing the original data, we find that factors C and E can be dropped. What type of $2^{k}$ design is left in the remaining variables?

$$\scriptstyle\begin{array}{rrrrrrr} \text { Run } & A & B & C & D & E=A B C & F=B C D & (\times 10) \\ \hline 1 & - & - & - & - & - & - & 6 \\ 2 & + & - & - & - & + & - & 10 \\ 3 & - & + & - & - & + & + & 32 \\ 4 & + & + & - & - & - & + & 60 \\ 5 & - & - & + & - & + & + & 4 \\ 6 & + & - & + & - & - & + & 15 \\ 7 & - & + & + & - & - & - & 26 \\ 8 & + & + & + & - & + & - & 60 \\ 9 & - & - & - & + & - & + & 8 \\ 10 & + & - & - & + & + & + & 12 \\ 11 & - & + & - & + & + & - & 34 \\ 12 & + & + & - & + & - & - & 60 \\ 13 & - & - & + & + & + & - & 16 \\ 14 & + & - & + & + & - & - & 5 \\ 15 & - & + & + & + & - & + & 37 \\ 16 & + & + & + & + & + & + & 52 \\ \hline \end{array}$$

Suppose the random variables X, Y, and Z have the joint probability density function f(x, y, z) = 8xyz for 0 < x < 1, 0 < y < 1, and 0 < z < 1. Determine the following: (a) Conditional probability distribution of X given that Y = 0.5 and Z = 0.8 (b) P(X < 0.5 | Y = 0.5, Z = 0.8)

a. List five values x such that x mod 7 = 0. b. List five values x such that x mod 5 = 2.

A card is drawn from a deck of cards numbered 6 - 19. Find each probability. P(14 or greater than 20)

Suppose that f(x) = 0.5x - 1 for 2 < x < 4. Determine the cumulative distribution function of the random variable.

Determine the range (possible values) of the random variable. A software program has 5000 lines of code. The random variable is the number of lines with a fatal error.

The function $f(x)=\frac{x}{x+1}$ is one-to-one, for x > -1, and has an inverse on that interval. a. Graph f, for x > -1. b. Find the inverse function $f^{-1}$ corresponding to the function graphed in part (a). Graph $f^{-1}$ on the same set of axes as in part (a). c. Evaluate the derivative of $f^{-1}$ at the point $\left(\frac{1}{2}, 1\right)$. A. Sketch the tangent lines on the graphs of $f$ and $f^{-1}$ at $\left(1 . \frac{1}{2}\right)$ and $\left(\frac{1}{2}, 1\right),$ respectively

Let G be the subgroup $\langle 3\rangle$ of $\mathbb{Z},$ and let N be the subgroup $\langle 15\rangle$. Find the order of 6 + N in the group G/N.

Perception & Psychophysics (July 1998) reported on a study of how people view three-dimensional objects projected onto a rotating two-dimensional image. Each in a sample of 25 university students viewed various depth-rotated objects (e.g., a hairbrush, a duck, and a shoe) until they recognized the object. The recognition exposure time-that is, the minimum time (in milliseconds) required for the subject to recognize the object-was recorded for each object In addition, each subject rated the "goodness of view" of the object on a numerical scale, with lower scale values corresponding to better views. The following table gives the correlation coefficient r between recognition exposure time and goodness of view for several different rotated objects:

$$\begin{matrix} \text{Object} & \text{r} & \text{t}\\ \text{Piano} & \text{.447} & \text{2.40}\\ \text{Bench} & \text{-057} & \text{.27}\\ \text{Motorbike} & \text{.619} & \text{3.78}\\ \text{Armchair} & \text{.294} & \text{1.47}\\ \text{Teapot} & \text{.949} & \text{14.50}\\ \end{matrix}$$

a. Interpret the value of r for each object. b. Calculate and interpret the value of $r^2$ for each object. c. The table also includes the t-value for testing the null hypothesis of no correlation (i.e., for testing $H_{0}: \beta_{1}=0$). Interpret these results using $x = .05.$

The Sports Journal (Winter 2004) published the results of a study conducted to assess the factors that affect the time allotted to sports news on local television news broadcasts. information on total time (in minutes) allotted to sports and on audience ratings of the TV news broadcast (measured on a 100-point scale) was obtained from a national sample of 163 news directors. A correlation analysis of the data yielded $r = .43$. Interpret the value of the correlation coefficient r.

A self-employed software engineer starts saving $300 a month in an index fund that returns 9$A(t)=40,000\left(1.0075^{12t}-1\right)$.

a. How much money will be in her account after 20 years and how fast is her account growing at this point?

b. How long will it take until the balance in her account reaches$100, 000, and how fast is her account growing at this point?

Recall that the substitution x = a sec $\theta$ implies either $x \geq a$ (in which case $0 \leq \theta<\pi / 2$ and $\tan \theta \geq 0$) or $x \leq-a$ (in which case $\pi / 2<\theta \leq \pi$ and $\tan \theta \leq 0)$. Show that

$$\int \frac{d x}{x \sqrt{x^{2}-1}}=\left\{\begin{array}{ll} \sec ^{-1} x+C & \text { if } x>1 \\ -\sec ^{-1} x+c & \text { if } x<-1 \end{array}\right.$$

A toy has two red LEDs $\left(\Delta V_{Open B}=1.6 \mathrm{V}\right)$, two green LEDs $\left(\Delta V_{\text {Open }} c=21 \vee\right)$, and two blue LEDs $\left(\Delta V_{Open B}=2.3 \mathrm{V}\right)$ connected in series. (a) What is the smallest number of 1.2-V rechargeable batteries needed to make all the LEDs glow? (b) How do you need to connect the batteries (in series or in parallel)? (c) For how many hours will the toy work with that number of batteries if the current through the LEDs is 20 mA and the total storage capacity of the batteries is 1200 mAh?

For each of the demand equations, find the rate of change of price p with respect to quantity q. What is the rate of change for the indicated values of q? $p=9 e^{-5 q / 750} ; q=300$