Describe the test statistic for the runs test when the sample sizes $n_1$ and $n_2$ are less than or equal to 20 and when either $n_1$ or $n_2$ is greater than 20.

Find the indefinite integral. $\int\left(2+x+2 x^{2}+e^{x}\right) d x$

(a) identify the claim and state $H_0$ and $H_a$, (b) decide which nonparametric test to use, (c) find the critical value(s), (d) find the test statistic, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim. The table shows the numbers of emails sent and the numbers of emails received in a week for a random sample of nine people. At $\alpha=0.01,$ can you conclude that there is a significant correlation between the number of emails sent and the number of emails received?

$$\begin{matrix} \text{Emails sent} & \text{30} & \text{30} & \text{25} & \text{26} & \text{24} & \text{18} & \text{18} & \text{25} & \text{28}\\ \text{Emails received} & \text{32} & \text{36} & \text{21} & \text{22} & \text{20} & \text{20} & \text{22} & \text{23} & \text{23}\\ \end{matrix}$$

Show that C(u, v) = uv is a copula. Why is it called “the independence copula”?

Test the Spearman rank correlation coefficient. The table shows the average hours worked per week and the numbers of on-the-job injuries for a random sample of U.S. construction companies in a recent year. At $\alpha=0.05$, can you conclude that there is a significant correlation between average hours worked and the number of on-the-job injuries?

$$\begin{matrix} \text{Hours worked} & \text{38} & \text{38} & \text{37} & \text{38} & \text{38} & \text{40} & \text{39} & \text{39} & \text{39} & \text{40} & \text{39} & \text{41}\\ \text{Injuries} & \text{11} & \text{11} & \text{9} & \text{10} & \text{10} & \text{17} & \text{15} & \text{14} & \text{14} & \text{16} & \text{15} & \text{17}\\ \text{Hours worked} & \text{41} & \text{42} & \text{41} & \text{41} & \text{41} & \text{42} & \text{42} & \text{42} & \text{42} & \text{41} & \text{41} & \text{39}\\ \text{Injuries} & \text{17} & \text{21} & \text{18} & \text{18} & \text{18} & \text{22} & \text{21} & \text{19} & \text{21} & \text{18} & \text{17} & \text{12}\\ \text{Hours worked} & \text{38} & \text{38} & \text{39} & \text{39} & \text{36} & \text{37} & \text{36} & \text{37} & \text{37} & \text{37} & \text{37} & \text{ }\\ \text{Injuries} & \text{12} & \text{11} & \text{13} & \text{12} & \text{6} & \text{6} & \text{6} & \text{6} & \text{7} & \text{8} & \text{7} & \text{ }\\ \end{matrix}$$

Let R be a ring with 1. Show that $(-1)^{2}=1$ in R.

Fill in the blanks. a. A function f has an absolute maximum at c if ____ for all x in the domains D of f. The number f(c) is called the ___ ___ ___ of f on D. b. A function f has a relative minimum at c if _____ for all values of x is some ____ ___ containing c.

A drawer of socks contains seven black socks, eight blue socks, and nine green socks. Two socks are chosen in the dark. a. What is the probability that they match? b. What is the probability that a black pair is chosen?

Find the Taylor series about the given point for each of the functions. In each case find the radius of convergence. $f(x)=\sin x^{2}$ about $x_{0}=0$

Either use an appropriate theorem to show that the given set, W, is a vector space, or find a specific example to the contrary.

$$\left\{\left[\begin{array}{l} a \\ b \\ c \\ d \end{array}\right]: \begin{array}{l} 3 a+b=c \\ a+b+2 c=2 d \end{array}\right\}$$

Find The critical value(s) and rejection region(s) for the type of t-test with level of significance $\alpha$ and sample size n. Left-tailed test, $\alpha=0.05, n=48$

Find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance $\alpha$. Two-tailed test, $n=31, \alpha=0.05$

When every collision between reactants leads to a reaction, what determines the rate at which the reaction occurs?

Fill in the details of the proof that the symmetric groups $S_{\Delta}$ and $S_{\Omega}$ are isomorphic if $|\Delta|=|\Omega|$ as follows: let $\theta: \Delta \rightarrow \Omega$ be a bijection. Define

$$\varphi: S_{\Delta} \rightarrow S_{\Omega} \quad \text { by } \quad \varphi(\sigma)=\theta \circ \sigma \circ \theta^{-1} \quad \text { for all } \sigma \in S_{\Delta}$$

and prove the following: (a) $\varphi$ is well defined, that is, if $\sigma$ is a permutation of $\Delta$ then $\theta \circ \sigma \circ \theta^{-1}$ is a permutation of $\Omega$. (b) $\varphi$ is a bijection from $S_{\Delta}$ onto $S_{\Omega}$. [Find a 2-sided inverse for $\varphi$. (c) $\varphi$ is a homomorphism, that is,

$$\varphi(\sigma \circ \tau)=\varphi(\sigma) \circ \varphi(\tau)$$