Optimus Company manufactures a variety of tools and industrial equipment. The company operates through three divisions. Each division is an investment center. Operating data for the Home Division for the year ended December 31, 2017, and relevant budget data are as follows.

	Actual	Comparison with Budget
Sales	$1,400,000	$100,000 favorable
Variable cost of goods sold	665,000	45,000 unfavorable
Variable selling and administrative expenses	125,000	25,000 unfavorable
Controllable fixed cost of goods sold	170,000	On target
Controllable fixed selling and administrative expenses	80,000	On target

Average operating assets for the year for the Home Division were $2,000,000 which was also the budgeted amount.

Instructions (a) Prepare a responsibility report (in thousands of dollars) for the Home Division.

Use the fact that the variance of a Poisson distribution is $\sigma^2=\mu$.

The mean snowfall in January for Bridgeport. Connecticut is 7.6 inches

a) Find the variance and standard deviation. Interpret the results.

b) Find the possibility that the snowfall in January for Bridgeport. Connecticut will exceed 12 inches,

Let x=age in years of a rural Quebec woman at the time of her first marriage. In the year 1941, the population variance of x was approximately $\sigma^{2}=5.1$. Suppose a recent study of age at first marriage for a random sample of 41 women in rural Quebec gave a sample variance $s^{2}=3.3$. Use a 5% level of significance to test the claim that the current variance is less than 5.1. Find a 90% confidence interval for the population variance. Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution?

A mathematics test of 100 multiple-choice questions is to be given to all freshmen entering a large university. Initially, in a pilot study the test was given to a random sample of 20 freshmen. Suppose that, for the population of all entering freshmen, the distribution of the number of correct answers would be normal with a variance of 250.

What is the probability that the sample variance would be less than 100?

What is the probability that the sample variance would be more than 500?

Determine if the matrix pairs are controllable.

$$A=\left[ \begin{array}{cccc}{0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {1} \\ {-2} & {-4.2} & {-4.8} & {-3.6}\end{array}\right], B=\left[ \begin{array}{r}{1} \\ {0} \\ {0} \\ {-1}\end{array}\right]$$

Find the mean, standard deviation, and variance of the data 13.5. 13.2, 12.1, 13.6, $13.3$.

Match each term on the left with its meaning on the right.

$$\begin{aligned} \text{superficial}\qquad \qquad \qquad\quad & \text{a. pertaining to the sole of the foot}\\ &\text{b. tail bone}\\ &\text{c. ventral cavity that contains heart, lungs, and associated structures}\\ &\text{d. toward the surface of the body (external)}\\ &\text{e. lying horizontal with face downward}\\ &\text{f. turning outward}\\ &\text{g. nearer to the center (trunk of the body)}\\ &\text{h. ventral cavity that contains digestive, reproductive, and excretory structures}\\ &\text{i. turning inward or inside out}\\ &\text{j. part of the spine known as the neck}\\ &\text{k. movement toward the median plane}\\ &\text{l. away from the head; toward the tail or lower part of a structure}\\ &\text{m. away from the surface of the body (internal)}\\ &\text{n. part of the spine known as the loin}\\ &\text{o. near the back of the body}\\ \end{aligned}$$

Match each term on the left with its meaning on the right.

$$\begin{aligned} \text{inversion}\qquad \qquad \qquad\quad & \text{a. pertaining to the sole of the foot}\\ &\text{b. tail bone}\\ &\text{c. ventral cavity that contains heart, lungs, and associated structures}\\ &\text{d. toward the surface of the body (external)}\\ &\text{e. lying horizontal with face downward}\\ &\text{f. turning outward}\\ &\text{g. nearer to the center (trunk of the body)}\\ &\text{h. ventral cavity that contains digestive, reproductive, and excretory structures}\\ &\text{i. turning inward or inside out}\\ &\text{j. part of the spine known as the neck}\\ &\text{k. movement toward the median plane}\\ &\text{l. away from the head; toward the tail or lower part of a structure}\\ &\text{m. away from the surface of the body (internal)}\\ &\text{n. part of the spine known as the loin}\\ &\text{o. near the back of the body}\\ \end{aligned}$$

Let $f ( x , y ) = \left( 2 / \theta ^ { 2 } \right) e ^ { - ( x + y ) / \theta } , 0 < x < y < \infty,$ zero elsewhere, be the joint pdf of the random variables X and Y. (a) Show that the mean and the variance of Y are, respectively, $3 \theta / 2$ and $5 \theta ^ { 2 } / 4$ (b) Show that $E ( Y | x ) = x + \theta.$ In accordance with the theory, the expected value of $X + \theta$ is that of Y , namely, $3 \theta / 2,$ and the variance of $X + \theta$ is less than that of Y . Show that the variance of $X + \theta$ is in fact $\theta ^ { 2 } / 4$

A random sample of 87 deer from a huge herd had a mean weight of 93.8 kg with a variance of 45.9 kg. a. Find the standard deviation of the sample. b. Find an unbiased estimation of the mean and variance of the entire herd from which the sample was taken.

Jams and Jellies, Inc., uses a standard cost system to track inventories and cost of goods sold. The blueberry factory that produces blueberry jams and jellies increased its standard product costs at the beginning of the fourth quarter of its operations for the current year. At the end of the year, the blueberry factory had a large favorable standard cost variance account balance. Buck, the manager of the blueberry factory, receives a large bonus based on the annual factory profits.

Buck has told Sheila, the factory accountant, to close the favorable variance account to Cost of Goods Sold. Sheila remembers that materials variance account balances should be allocated between the Inventory accounts and the Cost of Goods Sold account. However, Buck says that, although the total variance might seem material for the factory, in the grand scheme of the entire company the blueberry factory total variance is not material and Sheila should do as he has suggested.

Instructions

a. Why do you think Buck wants to close the entire variance balance to Cost of Goods Sold? If the variance account balance is closed to Cost of Goods Sold, will the ending finished goods inventory be overstated or understated?

Suppose that the variance for a set of data is zero. What can you say about the data?

Compute the variance and standard deviation for each collection of data. Round to the nearest tenth.

$$1,3,5,7,9,11$$

Is the mass of an atomic nucleus greater or less than the sum of the masses of the nucleons composing it? Why don't the nucleon masses add up to the total nuclear mass?