A random sample of the weekly supermarket bills for a number of families was observed and recorded in the table given. a. Find the mean bill and the standard deviation of the bills.b. Find unbiased estimates of the mean and variance of the population from which the data was taken.

$$\begin{matrix} Bill (\$) & \text{No. of families}\\ \text{70-79.99} & \text{27}\\ \text{80-89.99} & \text{32}\\ \text{90-99.99} & \text{48}\\ \text{100-109.99} & \text{25}\\ \text{110-119.99} & \text{37}\\ \text{120-129.99} & \text{21}\\ \text{130-139.99} & \text{18}\\ \text{140-149.99} & \text{7}\\ \end{matrix}$$

a. List the endocrine glands. b. How do the hypothalamus an the pituitary gland interact? c. Explain how the hypothalamus indirectly controls growth from infancy to adulthood.

Understand how X-linked inheritance differs from autosomal inheritance.

Let $V=\mathbb{R}^{5}$ with the standard basis and consider the vectors

$$v_{1}=(1,1,3,-2,3), v_{2}=(0,1,0,-1,0), v_{3}=(2,3,6,-5,6)$$

$$v_{4}=(0,3,1,-3,1), v_{5}=(2,-1,-1,-1,-1)$$

(a) Show that the reduced row echelon form of the matrix

$$A=\left( \begin{array}{rrrrr}{1} & {0} & {2} & {0} & {2} \\ {1} & {1} & {3} & {3} & {-1} \\ {3} & {0} & {6} & {1} & {-1} \\ {-2} & {-1} & {-5} & {-3} & {-1} \\ {3} & {0} & {6} & {1} & {-1}\end{array}\right)$$

whose columns are the coordinates of $v_{1}, v_{2}, v_{3}, v_{4}, v_{5}$ is the matrix

$$A^{\prime}=\left( \begin{array}{lllll}{1} & {0} & {2} & {0} & {2} \\ {0} & {1} & {1} & {0} & {18} \\ {0} & {0} & {0} & {1} & {-7} \\ {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0}\end{array}\right)$$

where the $1^{\text { st }}, 2^{\text { nd }}$ and $4^{\text { th }}$ columns are pivotal and the remaining two are nonpivotal. (b) Conclude that these vectors are linearly dependent, that the subspace $W$ spanned by $v_{1}, v_{2}, v_{3}, v_{4}, v_{5}$ is 3 -dimensional and that the vectors $v_{1}=(1,1,3,-2,3), \quad v_{2}=(0,1,0,-1,0) \quad$ and $\quad v_{4}=(0,3,1,-3,1)$ are a basis for $W$ (c) Conclude from (a) that the coefficients $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ of any linear relation

$$x_{1} v_{1}+x_{2} v_{2}+x_{3} v_{3}+x_{4} v_{4}+x_{5} v_{5}=0$$

satisfied by $v_{1}, v_{2}, v_{3}, v_{4}, v_{5}$ are given by the equations

$$x_{1}+2 x_{3}+2 x_{5}=0$$

$$x_{2}+x_{3}+18 x_{5}=0$$

$$x_{4}-7 x_{5}=0$$

Deduce that the $3^{\text { rd }}$ and $5^{\text { th }}$ variables, namely $x_{3}$ and $x_{5},$ corresponding to the non- pivotal columns of $A^{\prime},$ can be prescribed arbitrarily and the remaining variables are then uniquely determined as:

$$x_{1}=-2 x_{3}-2 x_{5}$$

$$x_{2}=-x_{3}-18 x_{5}$$

$$x_{4}=7 x_{5}$$

to give all the linear dependence relations satisfied by $v_{1}, v_{2}, v_{3}, v_{4}, v_{5} .$ In particular show that

$$-2 v_{1}-v_{2}+v_{3}=0$$

and

$$-2 v_{1}-18 v_{2}+7 v_{4}+v_{5}=0$$

corresponding to $\left(x_{3}=1, x_{5}=0\right)$ and $\left(x_{3}=0, x_{5}=1\right),$ respectively.

Find each sum or difference. Simplify if possible. $\dfrac{5}{11}+\dfrac{4}{11}$

Given that every solution of

$$\frac{d y}{d x}+y=2 x e^{-x}$$

may be written in the form $y=\left(x^{2}+c\right) e^{-x},$ for some choice of the arbitrary constant $c,$ solve the following initial-value problems:

$$\begin{aligned} \text { (a) } \frac{d y}{d x}+y &=2 x e^{-x} \\ y(0) &=2 \end{aligned}$$

$$\begin{aligned} \text { (b) } \frac{d y}{d x}+y &=2 x e^{-x} \\ y(-1) &=e+3 \end{aligned}$$

Sketch the graph of the function. Use as many as possible of the key features such as domain, relative extrema, inflection points, concavity, asymptotes, intercepts, cusps, and vertical tangents.

$$f(x)=\frac{5}{1+e^{-x}}$$

A physiologist believes that the hormone angiotensin II is important in regulating thirst. To investigate this belief, she randomly samples 16 rats from the vivarium of the drug company where she works and places them in individual cages with free access to food and water. After they have grown acclimated to their new “homes,” the experimenter measures the amount of water each rat drinks in a 20-minute period. Then she injects each animal intravenously with a known concentration (100 micrograms per kilogram) of angiotensin II. The rats are then put back into their home cages, and the amount each drinks for another 20-minute period is measured. The results are shown in the following table. Scores are in millilitres drunk per 20 minutes.

$$\begin{matrix} & \text{Before} & \text{After}\\\text{Subject} & \text{Injection} & \text{Injection}\\\text{1} & \text{1.2} & \text{11.3}\\\text{2} & \text{0.8} & \text{10.7}\\\text{3} & \text{0.5} & \text{10.3}\\\text{4} & \text{1.3} & \text{11.5}\\\text{5} & \text{0.6} & \text{9.6}\\\text{6} & \text{3.5} & \text{3.3}\\\text{7} & \text{0.7} & \text{10.5}\\\text{8} & \text{0.4} & \text{11.4}\\\text{9} & \text{1.1} & \text{12.0}\\\text{10} & \text{0.3} & \text{12.8}\\\text{11} & \text{0.6} & \text{11.4}\\\text{12} & \text{0.3} & \text{9.8}\\\text{13} & \text{0.5} & \text{10.6}\\\text{14} & \text{4.1} & \text{3.2}\\\text{15} & \text{0.4} & \text{12.1}\\\text{16} & \text{1.0} & \text{11.2}\\\end{matrix}$$

a. What is the non directional alternative hypothesis? b. What is the null hypothesis? c. Using $\alpha=0.05_{2 \text { tail }}$, what is your conclusion? Assume the injection itself had no effect on drinking behavior. d. What error might you be making by your conclusion in part c? e. To what population does your conclusion apply?

The table shows the lifetimes of a sample of electric light globes. Draw a cumulative frequency graph for the data and use it to estimate:

$$\begin{matrix} \text{Life (hours)} & \text{Number of globes}\\ {0 \leqslant l<500} & \text{5}\\ {500 \leqslant l<1000} & \text{17}\\ {1000 \leqslant l<2000} & \text{46}\\ {2000 \leqslant l<3000} & \text{79}\\ {3000 \leqslant l<4000} & \text{27}\\ {4000 \leqslant l<5000} & \text{4}\\ \end{matrix}$$

the median life of a globe

Add. Check that your answer is reasonable.

$$2.19+34.92$$

For each compound inequality, give the solution set in both interval and graph form. $x \leq 1$ or $x \leq 8$

A statistics professor conducts a study to investigate the relationship between the performance of his students on exams and their anxiety. Ten students from his class are selected for the experiment. Just before taking the final exam, the 10 students are given an anxiety questionnaire. Here are final exam and anxiety scores for the 10 students:

$$\begin{matrix}\text{Student No.} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6} & \text{7} & \text{8} & \text{9} & \text{10}\\\text{Anxiety} & \text{28} & \text{41} & \text{35} & \text{39} & \text{31} & \text{42} & \text{50} & \text{46} & \text{45} & \text{37}\\\text{Final Exam} & \text{82} & \text{58} & \text{63} & \text{89} & \text{92} & \text{64} & \text{55} & \text{70} & \text{51} & \text{72}\\\end{matrix}$$

a. On a piece of graph paper, construct a scatter plot of the paired scores. Use anxiety as the X variable. b. Describe the relationship shown in the graph. c. Assuming the relationship is linear, compute the value of Pearson r. d. Determine the least-squares regression line for predicting the final exam score, given the anxiety level. Should bY be positive or negative? Why? e. Draw the least-squares regression line of part d on the scatter plot of part a. f. Based on the data of the 10 students, if a student has an anxiety score of 38, what value would you predict for her final exam score? (Round to two decimal places.) g. Calculate the standard error of estimate for predicting final exam scores from anxiety scores.

A car moves around a 50-m-radius highway curve. The road,banked at $10^{\circ}$ relative to the horizontal, is wet and icy so that the coefficient of friction is approximately zero. At what speed should the car travel so that it makes the turn without slipping?

A leading toothpaste manufacturer advertises that, in a recent medical study, 70% of the people tested had brighter teeth after using its toothpaste (called Very Bright) as compared to using the leading competitor’s brand (called Brand X). The advertisement continues, “Therefore, use Very Bright and get brighter teeth.” In point of fact, the data upon which these statements were based were collected from a random sample of 10 employees from the manufacturer’s Pasadena plant. In the experiment, each employee used both toothpastes. Half of the employees used Brand X for 3 weeks, followed by Very Bright for the same time period. The other half used Very Bright first, followed by Brand X. A brightness test was given at the end of each 3-week period. Thus, there were two scores for each employee, one from the brightness test following the use of Brand X and one following the use of Very Bright. The following table shows the scores (the higher, the brighter):

$$\begin{matrix}\text{Subject} & \text{Very Bright} & \text{Brand X}\\\text{1} & \text{5} & \text{4}\\\text{2} & \text{4} & \text{3}\\\text{3} & \text{4} & \text{2}\\\text{4} & \text{2} & \text{3}\\\text{3} & \text{3} & \text{1}\\\text{6} & \text{4} & \text{1}\\\text{7} & \text{1} & \text{3}\\\text{8} & \text{3} & \text{4}\\\text{9} & \text{6} & \text{5}\\\text{10} & \text{6} & \text{4}\\\end{matrix}$$

a. What is the alternative hypothesis? Use a directional hypothesis. b. What is the null hypothesis? c. Using $\alpha=0.05_{1 \text { tail }}$, what do you conclude? d. What error might you be making by your conclusion in part c? e. To what population does your conclusion apply? f. Does the advertising seem misleading?