Ehrlich Co. began business on January 2. Salaries were paid to employees on the last day of each month, and social security tax, Medicare tax, and federal income tax were withheld in the required amounts. An employee who is hired in the middle of the month receives half the monthly salary for that month. All required payroll tax reports were filed, and the correct amount of payroll taxes was remitted by the company for the calendar year. Early in the following year, before the Wage and Tax Statements (Form W-2) could be prepared for distribution to employees and for filing with the Social Security Administration, the employees' earnings records were inadvertently destroyed.

None of the employees resigned or were discharged during the year, and there were no changes in salary rates. The social security tax was withheld at the rate of 6.0% and Medicare tax at the rate of 1.5% on salary. Data on dates of employment, salary rates, and employees' income taxes withheld, which are summarized as follows, were obtained from personnel records and payroll records:

Required:

1. Calculate the amounts to be reported on each employee's Wage and Tax Statement (Form W-2). Enter amounts to the nearest cent if required. Enter all amounts as positive numbers.

2. Calculate the following employer payroll taxes for the year: (a) social security; (b) Medicare; (c) state unemployment compensation at 5.4% on the first $10,000 of each employee's earnings; (d) federal unemployment compensation at 0.8% on the first $10,000 of each employee's earnings; (e) total. Round your answers to two decimal places.

Ehrlich Co. began business on January 2, 20Y8. Salaries were paid to employees on the last day of each month, and social security tax, Medicare tax, and federal income tax were withheld in the required amounts. An employee who is hired in the middle of the month receives half the monthly salary for that month. All required payroll tax reports were filed, and the correct amount of payroll taxes was remitted by the company for the calendar year. Early in 20Y9, before the Wage and Tax Statements (Form W-2) could be prepared for distribution to employees and for filing with the Social Security Administration, the employees' earnings records were inadvertently destroyed.

None of the employees resigned or were discharged during the year, and there were no changes in salary rates. The social security tax was withheld at the rate of 6.0% and Medicare tax at the rate of 1.5%. Data on dates of employment, salary rates, and employees’ income taxes withheld, which are summarized as follows, were obtained from personnel records and payroll records:

Required:

1.

Note: Round amounts to the nearest whole dollar and enter all amounts as positive values.

2. Calculate the following employer payroll taxes for the year: (a) social security; (b) Medicare; (c) state unemployment compensation at 5.4% on the first $10,000 of each employee’s earnings; (d) federal unemployment compensation at 0.6% on the first $10,000 of each employee’s earnings; (e) total.

Note: Round amounts to the nearest whole dollar and enter all amounts as positive values.

Calculate the amounts to be reported on each employee's Wage and Tax Statement (Form W-2) for 20Y8.

(10 pts) Suppose that when I drive to school, I encounter one traffic light on Lewis Road and one traffic light on Santa Rosa Rd. Let the random variable X = number of red lights that I encounter on Lewis and Y = number of red lights that I encounter on Santa Rosa. Suppose that the marginal distributions of X and Y are as shown in the following probability table:

X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0
Notice that E(X) = E(Y) = .5, and Var(X) = Var(Y) = .25.

a) Fill in the table in such a way that Corr(X,Y) = 1. Verify that indeed it checks out.
X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0


b) Fill in the table in such a way that Corr(X,Y) = -1. Verify that indeed it checks out.
X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0


c) Fill in the table in such a way that Corr(X,Y) = 0. Verify that indeed it checks out.
X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0

Consider the variable W=X+Y, representing the total number of red lights I encounter on my drive to school.
d) Calculate E(W)





e) For each of the cases in parts a), b) and c), calculate SD(W)

The thickness of a cardboard sheet has a normal distribution with a mean of 3 cm and a standard deviation of 0.1 cm. 60 of these sheets are stacked on top of each other to transport to the next operation in the process.

a) Determine the probability distribution for the total height of the stack of 60 cardboard sheets. Give the name of the distribution and its parameter(s).

b) The trailer used to transport the cardboards sheets has a capacity for a stack of height 181 cm. What is the probability that all 60 sheets won’t fit?

c) Upon further study, it is decided the thickness of the cardboard is better modeled by a gamma distribution with k = 900 and v = 300. Now what is the probability that the capacity of the trailer is exceeded? (An approximate probability is fine.)

Do you recall Dr. Suess' children's book GREEN EGGS AND HAM? In the book, Sam successfully convinces his prospect to become a consumer of these products--an excellent example of personal selling! For this DB you will first need to read the book or watch this video: https://www.youtube.com/watch?v=B32u4qrIVZQ After reading the book or viewing the video,

1) identify and discuss one factor that contributed to Sam’s success as a “salesperson.” Think about each step in the selling process.

2) Name one thing Sam did not do at all.

3) Using Sam as your salesperson, provide an example of something you can imagine Sam doing and saying to improve his sales presentation.

1. (10pt) Let V and W be a vector space over R. Show that V × W together with (v0,w0)+(v1,w1)=(v0 +v1,w0 +w1) for v0,v1 ∈V, w0,w1 ∈W

   and

   λ·(v,w)=(λ·v,λ·w) for λ∈R, v∈V, w∈W is a vector space over R.

2. (5pt)LetV beavectorspaceoverR,λ,μ∈R,andu,v∈V. Provethat (λ+μ)(u+v) = ((λu+λv)+μu)+μv.

   (In your proof, carefully refer which axioms of a vector space you use for every equality. Use brackets and refer to Axiom 2 if and when you change them.)

1. Determine if the following statements are true or false. If a statement is true, prove it in general, If a statement is false, provide a specific counterexample.

Let V and W be finite-dimensional vector spaces over field F, and let φ: V → W be a linear transformation.

A) If φ is injective, then dim(V) ≤ dim(W).

B) If dim(V) ≤ dim(W), then φ is injective.

C) If φ is surjective, then dim(V) ≥ dim(W).

D) If dim(V) ≥ dim(W), then φ is surjective.

E) If V = {0} , then φ is injective.

F) If dim(V) NOT= dim(W), then φ is not bijective.

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Use simulation to prove that when X ∼ N(0, 1), Z ∼ N(0, 1), Y = X3 + 10X +Z, we have V ar(X +Y ) = V ar(X) +V ar(Y ) + 2Cov(X, Y ) and V ar(X −Y ) = V ar(X) + V ar(Y ) − 2Cov(X, Y ).