Write the following systems using vectors and matrices.

$$\left\{\begin{array}{l} 2 x+5 y-z=2 \\ -x+2 y=-1 \\ x+4 z=0 \end{array}\right.$$

Explain how biodiversity loss can impact crop diversity.

Candidates A and B are running for office and 55% of the electorate favor candidate B. What is the probability that in a sample of size 100 at least one-half of those sampled will favor candidate A?

Suppose $X \sim N(4,2).$ What value of x is two standard deviations to the right of the mean?

A confidence interval for a population mean has a margin of error of 10.7. a. Obtain the length of the confidence interval. b. If the mean of the sample is 75.2, determine the confidence interval. c. Express the confidence interval in the form “point estimate $\pm$ margin of error."

If the level of significance is 0.05 and the p-value is 0.04, what conclusion can you draw?

When do you use an ANOVA test?

Let T: $\mathbf{P}^{2} \rightarrow \mathbf{R}^{2}$ be a linear transformation satisfying

$$T\left(x^{2}+1\right)=\left[\begin{array}{r} -1 \\ 3 \end{array}\right], \quad T(4 x+3)=\left[\begin{array}{l} 2 \\ 1 \end{array}\right]$$

. Find $T\left(2 x^{2}-4 x-1\right)$. (HINT: Express $2 x^{2}-4 x-1$ as a linear combination of $x^{2}+1$ and x+3.)

Let X and Y be discrete random variables and let g and h be functions such that the following identity holds:

$$P(X=x, Y=y)=g(x) h(y)$$

(a) Express P(X = x) in terms of g and h. (b) Express P(Y = y) in terms of g and h. (c) Show that

$$\left(\Sigma_{x} g(x)\right)\left(\Sigma_{y} h(y)\right)=1$$

(d) Show that X and Y are independent.

In what way does the overall magnitude of a stimulus affect the just-noticeable difference in the perception of that stimulus?

$$6 x ^ { 2 } - 10 x + 7$$

Cells in a hypertonic solution tend to: a. shrink due to water loss, b. swell due to water gain, c. stay the same size due to water moving into and out of the cell at the same rate, d. none of the above.

Find an example that meets the given specifications. A vector space V and a subset S that is almost a subspace: S contains O and is closed under scalar multiplication, but S is not closed under addition.

Suppose we have two decks of n cards, each numbered $1, \ldots, n.$ The two decks are shuffled and the cards are matched against each other. We say a match occurs at position i if the ith card from each deck has the same number. Let $S_{n}$ denote the number of matches. (a) Compute $E S_{n}$. (b) Compute Var $S_{n}$.