4(a) Suppose a particle P is moving in the plane so that its coordinates are given by P(x,y), where x = 4cos2t, y = 7sin2t.

x2 y2
(i) By finding a, b ∈ R such that a2 + b2 = 1, show that P is travelling on an elliptical

path. [10 marks] (ii) Let L(t) be the distance from P to the origin. Obtain an expression for L(t).[8 marks] (iii) How fast is the distance between P and the origin changing when t = π/8?[7 marks]

(b) A wire of length 100 centimeters is cut into two pieces. One piece is bent to form a square. The other piece is bent to form an equilateral triangle. Find the dimensions of the two pieces of wire so that the sum of the areas of the square and the triangle is minimized.(25marks)

A survey of high school girls classified them by two attributes: whether or not they participated in sports and whether or not they had one or more older brothers.

1. Use the following data to test the null hypothesis that these two attributes of classification are independent.
2. Suppose we do the below hypothesis test instead:

where,

p1-Propotion of young girls participated in sports of those who has a brother

p2-Propotion of young girls participated in sports of those who does not have a brother.

What would be the test statistic and the p value for this test?

There are two unequal life time mutually exclusive alternative investments, A and B with the following cash flows. Other investment opportunities exist at 14% minimum rate of return.

Project A

Project B

C: Cost, I:Income, L:Salvage

(1) Calculate the incremental NPV for this mutually exclusive project choice. Specifically, calculate NPV_B-A.

(2) Calculate the incremental ROR for this mutually exclusive project choice. Specifically, calculate ROR_A-B.

In this question, please enter your percentage answer as a whole number to two decimal places. For example, if your final answer is 7.5%, you would enter 7.5 in the text box, not 0.075.

38. Suppose a monopolist faces two groups of consumers. Group 1 has a demand given by P1 = 50−2Q1 and MR1 = 50−4Q1. Group 2 has a demand given by P2 = 40−Q2 and MR2 = 40−2Q2. The monopolist faces MC=AVC=ATC=$10 regardless of which group she supplies to. We can infer from the demand equations that in equilibrium Group __ is the inelastic group because the elasticity at that point is __ in absolute value than that of the other group.

    (a) 1; smaller. (b) 2; smaller.

    (c) 1; larger. (d) 2; larger.

I konw the answer is a, but I don't how to solve it.

You must show your work to get any credit.

Two projects have the following information:

Both projects required rate of return is 10%.

A) Calculate the NPV both projects. If they are mutually exclusive, which project would you pick?

B) Calculate the MIRR of both projects using the combo method and determine which project you would pick based solely on MIRR.

C) Calculate the cross-over rate

I have a group of 40 people that are suffering from headaches. I split them into two groups and gave the first group Aspirin and the second group Migraine medicine. There are 15 people in group one, 9 of them said that there headache where gone after taking the Aspirin. There are 25 people in group two that took the migraine medicine and 19 of them said that there headaches where gone. Is there significant evidence to say that the portion of people that felt better after taking the migraine medicine is great than the portion that took the Aspirin ?

40 people

Group one: 25 people 19 felt better

Group two: 15 people 9 felt better

A randomised controlled trial was conducted to assess the effectiveness of a new eye drop to reduce eye pain in patients suffering from a systemic condition affecting both eyes. Each patient received the prescribed dose of the new eye drop in one eye and the same dose of an existing drop in the other eye. The investigators randomised which eye was treated with the new eye drop and which received the existing eye drop. 45 patients were asked to rate the level of pain in each eye on a 0–10 scale with higher scores indicating higher levels of pain.The dataset is provided below.The dataset contains the following variables.

• ID: Participant ID number

• Pain_ctl = pain score in the eye receiving the existing drop

• Pain_new = pain score in the eye receiving the new drop

a) What are the study factor and the outcome factor?

b) State the null hypothesis and alternative hypothesis for the study.

c) Which statistical test would be appropriate to test the null hypothesis you have stated and why?

d) Use Stata to conduct the statistical test you stated in part (e) report the key findings and interpret them.Report all the relevant statistics and graphs required for checking the assumptions with interpretations

e) Write a brief report summarising all the results you have obtained and state your conclusion.

The future earnings, dividends, and common stock price of Callahan Technologies Inc. are expected to grow 8% per year. Callahan's common stock currently sells for $29.50 per share; its last dividend was $2.50; and it will pay a $2.70 dividend at the end of the current year.

Using the DCF approach, what is its cost of common equity? Round your answer to two decimal places. Do not round your intermediate calculations.
%

If the firm's beta is 1.60, the risk-free rate is 7%, and the average return on the market is 14%, what will be the firm's cost of common equity using the CAPM approach? Round your answer to two decimal places.
%

If the firm's bonds earn a return of 12%, based on the bond-yield-plus-risk-premium approach, what will be r_s? Use the midpoint of the risk premium range discussed in Section 10-5 in your calculations. Round your answer to two decimal places.
%

If you have equal confidence in the inputs used for the three approaches, what is your estimate of Callahan's cost of common equity? Round your answer to two decimal places. Do not round your intermediate calculations.
%

6.

The future earnings, dividends, and common stock price of Callahan Technologies Inc. are expected to grow 8% per year. Callahan's common stock currently sells for $24.50 per share; its last dividend was $2.00; and it will pay a $2.16 dividend at the end of the current year.

A.

Using the DCF approach, what is its cost of common equity? Do not round intermediate calculations. Round your answer to two decimal places.

B.

If the firm's beta is 1.3, the risk-free rate is 7%, and the average return on the market is 13%, what will be the firm's cost of common equity using the CAPM approach? Round your answer to two decimal places.

C.

If the firm's bonds earn a return of 11%, based on the bond-yield-plus-risk-premium approach, what will be rs? Use the midpoint of the risk premium range discussed in Section 10-5 in your calculations. Round your answer to two decimal places.

D.

If you have equal confidence in the inputs used for the three approaches, what is your estimate of Callahan's cost of common equity? Do not round intermediate calculations. Round your answer to two decimal places.