In 2019, Pittsburgh Steelers starting quarterback Ben Roethlisberger suffered a season-ending elbow injury in Week 2. The team struggled to find a quarterback until Week 12, when undrafted rookie Devlin “Duck” Hodges entered a game against the Cincinnati Bengals at halftime and led the Steelers to a surprising comeback victory. His performance earned him the starting job for the rest of the season and engendered a flurry of media speculation about his potential as a future quarterback for the Steelers.

Assume quarterbacks can either be good, average, or bad. Because Hodges was undrafted, it’s fair to say that coaches and fans had low expectations for him. Assume the initial probability distribution for Hodges was:

P(Good) =0.1

P(Average)=0.2

P(Bad)= 0.7

Also assume quarterbacks can either have strong performances or weak performances (there are only two types of performances, strong or weak). The chance of a good quarterback having a strong performance is 80%, the chance of an average QB having a strong performance is 50%, and the chance of a bad QB having a strong performance is 30%. We have:

P(S | G) =0.8

P(S | A)=0.5

P(S | B)= 0.3

From ESPN.com, Hodges turned in the following performances during the last 6 weeks of the season:

Use Bayes’s Rule to fill in the following table with the week-by-week probabilities of Duck being good, average, or bad.

In 2019, Pittsburgh Steelers starting quarterback Ben Roethlisberger suffered a season-ending elbow injury in Week 2. The team struggled to find a quarterback until Week 12, when undrafted rookie Devlin “Duck” Hodges entered a game against the Cincinnati Bengals at halftime and led the Steelers to a surprising comeback victory. His performance earned him the starting job for the rest of the season and engendered a flurry of media speculation about his potential as a future quarterback for the Steelers.

Assume quarterbacks can either be good, average, or bad. Because Hodges was undrafted, it’s fair to say that coaches and fans had low expectations for him. Assume the initial probability distribution for Hodges was:

P(Good) =0.1

P(Average)=0.2

P(Bad)= 0.7

Also assume quarterbacks can either have strong performances or weak performances (there are only two types of performances, strong or weak). The chance of a good quarterback having a strong performance is 80%, the chance of an average QB having a strong performance is 50%, and the chance of a bad QB having a strong performance is 30%. We have:

P(S | G) =0.8

P(S | A)=0.5

P(S | B)= 0.3

From ESPN.com, Hodges turned in the following performances during the last 6 weeks of the season:

Use Bayes’s Rule to fill in the following table with the week-by-week probabilities of Duck being good, average, or bad.

We revisit Example 3.6 in Chapter 3’s lecture note, which shows the relationship between the price of a diamond and its carat.

Now we assume that the relationship is of the form P = a 0 e a 1 C .

a) (10pt) Discuss how to estimate a 0 and a 1 using linear regression technique.

b) (10pt) Using the method discussed in a), by Excel, estimate a 0 and a 1 , determine a CER equation to fit these data, and plot the data points and the associated CER curve.

c) (5pt) Using this method, what will be the price for a 1.2 carat diamond?

d) (10pt) Compute the standard error of this method.

e) (10pt) Using the coefficient obtained on page 31 of Chapter 3’s lecture note, compute the standard error of the method assuming the power form P = a 0 C a 1 , and compare it with the result in d). (Note that “SE = 0.1136" on the page is not an answer for this question.)

Research Scenario: Does distraction and/or amount of details affect the ability of people to make good decisions? In this fictitious scenario, researchers used a mixed design. Thirty participants were split into two groups – No Distraction or Distraction (n=15 per group). All participants were given TWO scenarios based on amount of details (4 or 12), and were asked to make an objective decision at the end of each scenario. Objective decision was the dependent variable and was quantified numerically using an interval scale of measurement.

Assume the data is parametric. Select and conduct the most appropriate statistical test to determine whether distraction and/or amount of details affect people’s ability to make good decisions.

(a) Chau’s electric circuit is a simple electronic circuit that can exhibit chaotic behaviour. The voltages x(t) and y(t), and current z(t), across components in the circuit can be investigated using the Matlab command

[t,xyz] = ode45(@ChuaFunction,[-10 100],[0.7 0.2 0.3]);

and the function:

1 function dxyzdt = ChuaFunction(~,xyz)

2 % xyz(1) = X, xyz(2) = Y, xyz(3) = Z

3 4 dxdt = 15.6*(xyz(2) - xyz(1) + 2*tanh(xyz(1)));

5 dydt = xyz(1) - xyz(2) + xyz(3);

6 dzdt = -28*xyz(2); 7

8 dxyzdt = [dxdt dydt dzdt]’;

9 end

(i) What is the differential equation involving x˙(t)? Here a dot represents differentiation with respect to time t.

(ii) What is the initial condition for the variable y(t)?

(iii) What does the apostrophe after the square brackets in line 8 of the function signify and why is the apostrophe needed here?

(b) For a given function u(t), explain how the derivative of u(t) with respect to t can be approximated on a uniform grid with grid spacing ∆t, using the one-sided forward difference approximation

du/dt ≈ Ui+1 − Ui/ ∆t ,

where ui = u(ti). You should include a suitable diagram explaining your answer

(c) Using the one-sided forward difference approximation from part (b) and Euler’s method, calculate the approximate solution to the initial value problem

du/dt + t cos(u) = 0, subject to u(0) = −0.2,

at t = 0.4, on a uniform grid with spacing ∆t = 0.1.

Particles of charge -60 E-6 C, +40 E-6 C, and – 95 E-6 C are placed along the x-axis at 0.2 m, 0.4 m and 0.6 m, respectively. (a) Calculate the magnitude of the net electric field x = 0.3 m. (b) Calculate the magnitude of the net force on the +40 E-6 C charge

Better Mousetraps has developed a new trap. It can go into production for an initial investment in equipment of $5.4 million. The equipment will be depreciated straight line over 6 years to a value of zero, but in fact it can be sold after 6 years for $584,000. The firm believes that working capital at each date must be maintained at a level of 10% of next year’s forecast sales. The firm estimates production costs equal to $1.30 per trap and believes that the traps can be sold for $5 each. Sales forecasts are given in the following table. The project will come to an end in 6 years, when the trap becomes technologically obsolete. The firm’s tax bracket is 35%, and the required rate of return on the project is 10%.

a. What is project NPV? (Negative amount should be indicated by a minus sign. Do not round intermediate calculations. Enter your answer in millions rounded to 4 decimal places.)

b. By how much would NPV increase if the firm depreciated its investment using the 5-year MACRS schedule?

PRACTICE ANOTHER

Suppose that a category of world class runners are known to run a marathon (26 miles) in an average of 149 minutes with a standard deviation of 12 minutes. Consider 49 of the races.

Let X = the average of the 49 races.

Find the 70th percentile for the average of these 49 marathons. (Round your answer to two decimal places.)

A random sample of six cars from a particular model year had the following fuel consumption figures ( in miles per gallon). find the 98% confidence interval for the true mean fuel consumption for cars of this model year.

Sample Data: 19.3 , 18.5 , 18.4 , 20.9 , 20.7 , 18.7

Left Endpoint _____

Right Endpoint _____