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

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

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

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 $\theta ^ { 2 } / 4$

State the probability density function for an exponential random variable with Variance $\frac{1}{4}$ .

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

$$5,5,5,5,5,5,5$$

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

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?

A producer of machine parts claimed that the diameters of the connector rods produced by his plant had a variance of at most .03 inch$^2$. A random sample of 15 connector rods from his plant produced a sample mean and variance of .55 inch and .053 inch$^2$, respectively. a. Is there sufficient evidence to reject his claim at the $\alpha$=.05 level of significance? b. Find a 95% confidence interval for the variance of the rod diameters.

Stoker Corporation uses fixed overhead at the rate of $0.50 per unit. For May, budgeted fixed overhead was$403,000. The production volume variance amounted to $3,000 unfavorable, and the price variance was$10,000 unfavorable.

Required
What was the budgeted volume in units for May?

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?

Let $X_1,X_2, . . . , X_n$ denote a random sample from a normal distribution with mean zero and variance $\theta , 0 < \theta < \infty$ Show that $\sum _ { 1 } ^ { n } X _ { i } ^ { 2 } / n$ is an unbiased estimator of $\theta$ and has variance $2 \theta ^ { 2 } / n$

Determine if the matrix pairs are controllable.

$$A=\left[ \begin{array}{cccc}{0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {1} \\ {-1} & {-13} & {-12.2} & {-1.5}\end{array}\right], B=\left[ \begin{array}{r}{1} \\ {0} \\ {0} \\ {-1}\end{array}\right]$$

Determine if the matrix pairs are controllable.

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