In testing a new drug, we obtained the following results:

Run the ANOVA and fill in the summary table with the results obtained:

(Report P-value & F-ratio accurate to 3 decimal places and all other values accurate to 2 decimal places.)

What conclusion can be drawn at the 0.01 significance level?

A) The various drugs have results that are statistically different.

B)The various drugs do not have results that are statistically different.

Assume that the Poisson process X = {X(t) : t ≥ 0} describes students’ arrivals at the library with intensity λ = 4 per hour. Given that the tenth student arrived exactly at the end of fourth hour, or W₁₀ = 4, find:

1. E [W₁|W₁₀ = 4]

2. E [W₉ − W₁|W₁₀ = 4].

Hint: Suppose that X {X(t) : t ≥ 0} is a Poisson process with rate λ > 0 and its arrival times are defined for any natural k as W_k = min[t ≥ 0 : X(t) = k] (1) Then for any natural m, the inter-arrival times, {T₁ = W₁, T₂ = W₂ − W₁, . . . , T_m = W_m − W_m−1} are independent variables with the common exponential distribution, f^T(t) = λ · e −λ·t for t > 0.

1. Consider the following table:

          

   Labor	Output	Marginal Product
   0	0	?
   10	100	?
   20	180	?
   30	240	?
   40	280	?

   Based on the table above, if labor increases from 20 to 30, then marginal product of the 30^th worker is:

   		10
   		8
   		6
   		4

2 points   

QUESTION 2

1. Suppose the long run production function is given by: Q = 4*L +2K². Marginal product of labor (MP_L) = 4 and wage is $10. Marginal product of capital (MP_K) = 4K and price of capital (K) is $10. Consider the allocation labor (L) = 10 and capital (K) = 2. Based on information, the MRTS is equal to

   		4
   		2.5
   		1
   		0.5

2 points   

QUESTION 3

1. The market supply of labor does NOT depend on:

   		non-monetary benefits.
   		working conditions.
   		mobility.
   		technology.

2 points   

QUESTION 4

1. In a perfectly competitive product market,

   		Price > MR
   		Price < MR.
   		Price = ME.
   		Price = MR.

2 points   

QUESTION 5

1. The marginal product for labor is given (MP) = 3 – 0.02*L; price of the product is $100 and wage = 200.  Based on information above, the marginal product of labor at the optimal level of employment is

   		$3
   		$2
   		$1.5
   		$1

2 points   

QUESTION 6

1. If the labor elasticity of output is 0.5 and the capital elasticity of output is 0.9, then the production function exhibits

   		constant returns to scale.
   		economies of scale.
   		diseconomies of scale.
   		diminishing returns.

2 points   

QUESTION 7

1. Suppose the long run production function is given by: Q = 4*L +2K². Marginal product of labor (MP_L) = 4 and wage is $10. Marginal product of capital (MP_K) = 4K and price of capital (K) is $10. Consider the allocation labor (L) = 10 and capital (K) = 2. Based on information, the MRTS is equal to

   		4
   		2.5
   		1
   		0.5

2 points   

QUESTION 8

1. If the demand for product increases,

   		labor demand increases.
   		labor demand decreases.
   		labor supply decreases.
   		labor supply increases.

2 points   

QUESTION 9

1. Suppose a firm is operating in both a perfectly competitive product market and perfectly labor market. The firm’s short run production is Q = L²; where Q is output and L is labor, expressed in millions. Marginal product of labor (MP_L) = 2L and wage is 10. The price of the product is $ 2. Based on information, the short run optimal level of employment is

   		4 million
   		2.5 million
   		5 million
   		0.4 million

2 points   

QUESTION 10

1. Consider the following production function: Q = KL where Q = output, L = labor and K = capital. The marginal product of labor is given by MP_L = K while the marginal product of capital is given by MP_K = L.   If L = 10 and K= 5, the marginal product of capital is

   		2
   		5
   		10
   		50

2 points   

QUESTION 11

1. At the market clearing wage,

   		labor supplied = labor demanded
   		labor supplied > labor demanded
   		labor supplied < labor demanded
   		None of these is true

You are the head of project selection for Broken Arrow Records (BAR). Your team is considering three new projects, each with a unique sound and style. Based on past history, management requires a 20% rate of return. Additionally, they have allocated $1 million toward the production of these albums. Finally, management wants you to find new talent without taking risks. So, give the following weights to projects;

New Artist = 10, Risk = 6, Genre = 3, Diversity = 2

Given the following information about each project, prioritize each project. That is, put them in order of which BAR should do first, second, and third; money permitting, of course.

Note: You will use the Project Selection Matrix, the Payback Period, and the NPV to make your decision.

Project: Time Fades Away

            New Artist: 10

            Risk: -10

Genre: 7

            Diversity: 3

Project: Tears in the Rain

            New Artist: 5

            Risk: -5

Genre: 9

            Diversity: 2

Project: On the Beach

            New Artist: 2

            Risk: -2

Genre: 3

            Diversity: 2

SHOW EXCEL FORMULAS

What does the following pseudocode axiom mean, in English?

1. ( aStack.push( item ) ).pop() = aStack

2. aList.getLength() = ( aList.insert( i, item ) ).getLength() - 1

3. ( aList.insert( i, item ) ).remove( i ) = aList

4. aList.getEntry( i ) = ( aList.insert( i, item ) ).getEntry( i+1 )

Using Matlab Simulink, find Fourier transform of the following signal;

?(?) = 2 + ∑
1 ?
sin (20???)
4
?=1
.

Set simulation stop time = 20 seconds, sample time = (1/1024) seconds, buffer size =1024, and frequency range in Hz for the vector scope is −100 ≤ ? ≤ 100

By making the following substitutions

• x becomes θ
• v becomes ω
• a becomes α
• F becomes τ
• m becomes I

we get a whole new physics that tells how rotating objects behave. Everything we have done all semester flips over to explain rotational physics

1. The moment of inertia is the quantity that replaces mass in all the old formulas. Not only is the mass of an object important, but also how that mass is distributed.

Find the moment of inertia of a 4 meter long stick with a mass of 23 kg, if it is spun about the center of the stick

I_Sphere = 2/5 MR²

I_Cylinder = 1/2 MR²

I_Ring = MR²

I_{Stick thru center} = 1/12 ML²

I_{Stick thru end} = 1/3 ML²

2. An ice skater with a moment of inertia of 10 kg m²spinning at 14 rad/s extends her arms, thereby changing her moment of inertia to 26 kg m^2. Find the new angular velocity.

Hint: conserve angular momentum!

3. Find the rotational kinetic energy of a spinning (not rolling) bowling ball that has a mass of 10 kg and a radius of 0.17 m moving at 12 m/s.

(Fun fact: How can this problem be done if r isn't given?)

v = rω

I_Sphere = 2/5 MR²

I_Cylinder = 1/2 MR²

I_Ring = MR²

I_{Stick thru center} = 1/12 ML²

I_{Stick thru end} = 1/3 ML^2

Recall: when rolling, the ball is both moving forward and rotating, so the total KE = the linear KE + the rotational KE

4. Find the total kinetic energy of a rolling bowling ball that has a mass of 8 kg and a radius of 0.19 m moving at 16 m/s.

v = rω

I_Sphere = 2/5 MR²

I_Cylinder = 1/2 MR²

I_Ring = MR²

I_{Stick thru center} = 1/12 ML²

I_{Stick thru end} = 1/3 ML²

Recall: E₁ = E₂

5. Find the height a rolling bowling ball that has a mass of 4 kg and a radius of 0.15 m moving at 7 m/s can roll up a hill.

v = rω

I_Sphere = 2/5 MR²

I_Cylinder = 1/2 MR²

I_Ring = MR²

I_{Stick thru center} = 1/12 ML²

I_{Stick thru end} = 1/3 ML^2

Hint: force at a distance is torque

6. A coke can is suspended by a string from the tab so that it spins with a vertical axis. A 17 N perpendicular force at the edge causes rotation. Find the angular acceleration if the can has a radius of 5 cm and a mass of 929 grams.

Hint: force at a distance is torque

I_Sphere = 2/5 MR²

I_Cylinder = 1/2 MR²

I_Ring = MR²

I_{Stick thru center} = 1/12 ML²

I_{Stick thru end} = 1/3 ML²

Complete the R code using Rstudio so that it calculates and returns the estimates of β, the intercept and regression weight of the logistic regression of approximate GPA on Rouder-Srinivasan preference.

## Data
Preference <- c(  0,   0,   0,   0,   0,   1,   1,   1,   1)  # 0: Rouder; 1: Srinivasan
GPA        <- c(2.0, 2.5, 3.0, 3.5, 4.0, 2.5, 3.0, 3.5, 4.0)
Count      <- c(  4,   5,  21,  22,   8,   2,   1,   4,   7)

# Define the deviance function
deviance <- function(beta) {
   ... complete this ...
}

## Test the function
deviance(c(0,1))

## Estimate
optim(c(0, 1), deviance)$par

solve by determinants

a.x+y+z=0

3x-y+2z=-1

2x+3y+3z=-5

b. x+2z=1

2x-3y=3

y+z=1

c. x+y+z=10

3x-y=0

3y-2z=-3

d. -8x+5z=-19

-7x+5y=4

-2y+3z=3

e. -x+2y+z-5=0

3x-y-z+7=0

-2x+4y+2z-10=0

f. 1/x+1/y+1/z=12

4/x-3/y=0

2/y-1/z=3

Do the following problems:

a. Use mathematical induction to show that 1 + 2 + 22 +···+ 2n = 2n+1 − 1 for all non-negative integers n.
b. A coin is weighted so that P(H) = 2/3 and P(T) = 1/3. The coin is tossed 4 times. Let the random variable X denote the number of heads that appear. (x) Find the distribution of X; (xx) Find the expectation E(X).
c. Show a derivation of Bayes’ Theorem