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?

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?

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 $\frac{1}{4}$ .

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.

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.

Are one-way analysis of variance tests left-tailed, right-tailed, or two-tailed?

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,

$\overline{\text{x}}_1$ and $S^2_1$ are the sample mean and sample variance from a population with mean $μ_1$ and variance $σ^2_1$. Similarly, $\overline{\text{x}}_2$ and $S^2_2$ are are the sample mean and sample variance from a second independent population with mean $μ_2$ and variance $σ^2_2$. The sample sizes are $n_1$ and $n_2$, respectively. a. Show that $\overline{\text{x}}_1$ - $\overline{\text{x}}_2$ is an unbiased estimator of $μ_1 - μ_2$. b. Find the standard error of $\overline{\text{x}}_1$ - $\overline{\text{x}}_2$. How could you estimate the standard error? c. Suppose that both populations have the same variance; that is, $σ^2_1 = σ^2_2 = σ^2$. Show that $S^2_p = (n_1 - 1)S^2_1 + (n_2 - 1)S^2_2/n_1 + n_2 - 2$

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.

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.

Find the mean, variance, and standard deviation of the binomial distribution with the given value of n and p.

n = 124, p = 0.26

Find the energy to remove (a) one proton and (b) one neutron from a ${ }^{32} \mathrm{~S}$ nucleus. (c) Explain the difference between your answers to parts (a) and (b).

Suppose that X has a gamma distribution with $\lambda=3$ and r = 6. Determine the mean and variance of X.