Use a sign test to test the claim by doing the following. (a) Identify the claim and state $H_0$ and $H_a$. (b) Find the critical value. (c) Find the test statistic. (d) Decide whether to reject or fail to reject the null hypothesis. (e) Interpret the decision in the context of the original claim. A store manager claims that the median number of customers per day is no more than 650. The numbers of customers per day for 17 randomly selected days are listed below. At $\alpha=0.01$, can you reject the manager’s claim?

$$\begin{matrix} \text{675 665 601 642 554 653 639 650 645}\\ \text{550 677 569 650 660 682 689 590}\\ \end{matrix}$$

identify the claim and state $H_0$ and $H_a$.

Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal. The well-being index is a way to measure how people are faring physically, emotionally, socially, and professionally, as well as to rate the overall quality of their lives and their outlooks, for the future. The table shows the well-being index scores for a sample of states from four regions of the United States. At $\alpha=0.10$, can you reject the claim that the mean score is the same for all regions?

$$\begin{matrix} \text{Northeast} & \text{Midwest} & \text{South} & \text{West}\\ \text{61.7} & \text{61.6} & \text{61.0} & \text{64.0}\\ \text{63.6} & \text{61.8} & \text{60.8} & \text{63.0}\\ \text{62.6} & \text{61.4} & \text{63.1} & \text{63.5}\\ \text{61.8} & \text{63.2} & \text{62.3} & \text{63.2}\\ \text{62.1} & \text{61.7} & \text{60.5} & \text{61.8}\\ \text{63.5} & \text{62.9} & \text{62.0} & \text{62.6}\\ \text{ } & \text{62.9} & \text{61.3} & \text{62.5}\\ \text{ } & \text{63.7} & \text{60.5} & \text{62.8}\\ \text{ } & \text{ } & \text{62.3} & \text{62.5}\\ \text{ } & \text{ } & \text{61.5} & \text{ }\\ \text{ } & \text{ } & \text{63.1} & \text{ }\\ \end{matrix}$$

If the null space of an $8 \times 5$ matrix A is 3-dimensional, what is the dimension of the row space of A?

Let R be the collection of sequences $\left(a_{1}, a_{2}, a_{3}, \ldots\right)$ of integers $a_{1}, a_{2}, a_{3}, \ldots$ where all but finitely many of the $a_{i}$ are 0 (called the direct sum of infinitely many copies of $\mathbb{Z}$). Prove that R is a ring under componentwise addition and multiplication which does not have an identity.

Use the data to (a) find the coefficient of determination $r^2$ and interpret the result, and (b) find the standard error of estimate $s_e$, and interpret the result. The U.S. voting age populations (in millions) and the number of ballots cast (in millions) for the highest office in federal elections for nine nonpresidential election years are shown in the table. The equation of the regression line is $\hat{y}=0.276 x+18.881$

$$\begin{matrix} \text{Voting age population, x} & \text{166.0} & \text{177.9} & \text{186.2} & \text{195.3} & \text{205.3}\\ \text{Ballots cast in federal} & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{elections, y} & \text{67.6} & \text{65.0} & \text{67.9} & \text{75.1} & \text{72.5}\\ \text{Voting age population, x} & \text{215.5} & \text{225.5} & \text{236.0} & \text{245.7} & \text{ }\\ \text{Ballots cast in federal} & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{elections, y} & \text{78.4} & \text{83.8} & \text{89.1} & \text{81.7} & \text{ }\\ \end{matrix}$$

Let f be a constant function-that is, let f(x)=c, where c is some real number. Show that every number a gives rise to an absolute maximum and, at the same time, an absolute minimum of f.

A truck gets 600/x mpg when driven at a constant speed of x mph (between 50 and 70 mph). If the price of fuel is $3/gallon and the driver is paid$18/hour, at what speed between 50 and 70 mph is it most economical to drive?

(a) identify the claim and state $H_0$ and $H_a$. (b) decide whether to use a Wilcoxon signed-rank test or a Wilcoxon rank sum test. (c) find the critical value(s). (d) find the test statistic. (e) decide whether to reject or fail to reject the null hypothesis. (f) interpret the decision in the context of the original claim. A physician wants to determine whether an experimental medication affects an individual’s heart rate. The physician randomly selects 15 patients and measures the heart rate of each. The subjects then take the medication and have their heart rates measured after one hour. The table shows the results. At $\alpha=0.05$, can the physician conclude that the experimental medication affects an individual’s heart rate?

$$\begin{matrix} \text{Patient} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6} & \text{7} & \text{8}\\ \text{Heart rate (before)} & \text{72} & \text{81} & \text{75} & \text{76} & \text{79} & \text{74} & \text{65} & \text{67}\\ \text{Heart rate (after)} & \text{73} & \text{80} & \text{75} & \text{79} & \text{74} & \text{76} & \text{73} & \text{67}\\ \text{Patient} & \text{9} & \text{10} & \text{11} & \text{12} & \text{13} & \text{14} & \text{15} & \text{ }\\ \text{Heart rate (before)} & \text{76} & \text{83} & \text{66} & \text{75} & \text{76} & \text{78} & \text{68} & \text{ }\\ \text{Heart rate (after)} & \text{74} & \text{77} & \text{70} & \text{77} & \text{76} & \text{75} & \text{74} & \text{ }\\ \end{matrix}$$

Determine the intervals where the function $f(x)=\frac{\ln x}{x}$ is increasing and where it is decreasing.

Use logarithmic differentiation to find the derivative of the function. $y=x^{\ln x}$

Universal Instruments found that the monthly demand for its new line of Galaxy Home Computers r months after placing the line on the market was given by $D(t)=2000-1500 e^{-0.05 t}$ (t>0) Graph this function and answer the following question: What is the demand after 1 month? After 1 year? After 2 years? After 5 years?

For what value of c is the vector $\mathbf{v}=\langle 2,-5, c\rangle$ orthogonal to $\mathbf{w}=\langle 3,2,9\rangle$?

Account for the increase in reaction rate brought about by a catalyst.

Show that the group of rigid motions of a cube is isomorphic to $S_{4}$. [This group acts on the set of four pairs of opposite vertices.]