The strength Y of a product has the Weibull distribution with pdf;

?(?) = {????−1?(−???),    ??? ? > 0, ? > 0 0, ?????ℎ???

If ? is unknown and ? is known. The strength of a random sample of n products are, ?1,?2,…,??.

a) Find the form of the most powerful test of the null hypothesis ? = 0.5 against the alternative hypothesis ? = 1.0.

b) Show that ? ? ? has an exponential distribution.

c) Use ?2 tables to find the critical region of the most powerful test at the 10% level when n=20. [Assume the result that if ?1,?2,…,?? are independent, each with an exponential distribution, mean ?, then 2?∑?? has the ?2? 2 distribution.]

Use the table for the​ question(s) below.

Consider the following three individuals portfolios consisting of investments in four​ stocks:

Assuming that the risk-free rate is​ 4% and the expected return on the market is​ 12%, then required return on​ Peter's Portfolio is closest​ to:

A.

​8%

B.

​12%

C.

​9%

D.

​10%

A portfolio manager is holding the following investments: Stock Amount Invested Beta X $12 million 1.4 Y 25 million 1.0 Z 40 million 0.6 The manager plans to sell his holdings of Stock Y. The money from the sale will be used to purchase another $15 million of Stock X and 10 million of Stock Z. The risk-free rate is 5 percent and the market risk premium is 5.5 percent. How many percentage points higher will the required return on the portfolio be after he completes this transaction?

In a study on a blood disease, the normal distribution of hemoglobin values and its arithmetic mean is 12.5 and its standard deviation is 1.0. Accordingly, this type of patients:

a) What is the probability that hemoglobin values ​​in the blood will be between 11.5 and 13.0? P (11.5 <X≤ 13.0) =?

b) What symmetrical limits are the hemoglobin values ​​of 80% of patients relative to the mean?

c) What is the hemoglobin value of the patient who has hemoglobin higher than 70% of the patients?

d) How many percent of patients have hemoglobin value less than 12.2? P (X <12.2) =?

A stock's returns have the following distribution:

Assume the risk-free rate is 2%. Calculate the stock's expected return, standard deviation, coefficient of variation, and Sharpe ratio. Do not round intermediate calculations. Round your answers to two decimal places.

Stock's expected return:   %

Standard deviation:   %

Coefficient of variation:

Sharpe ratio:

Answer the following question based on the following information: An investor is considering the following three stocks to invest. Suppose that the current T-bill rate is 5% and the market rate of return is 13%.

Stock A Stock B Stock C Beta

1.3 1.0    0.7

Suppose that the investor has invested $100,000 in stock A and $100,000 in stock B. How much should he/she invest in stock C so that he/she can expect 14% rate of return from the portfolio?

Group of answer choices

$5,124.76

$9.090.91

$0.00

$12,564.94

$11,764.71

We will test whether the number of hours spent on practice is the same for football players as for basketball players.  A sample of 9 basketball players averages 2 hours per day of practice with a sample standard deviation of 0.866. A sample of 16 football players averages 3.2 hours per day of practice with a sample standard deviation of 1.0. Each population has a normal distribution. Use a 2 tail test. Are the numbers equal or not using alpha equals 0.05?

What is the simplified degrees of freedom?

What are the rejection regions?

What is the absolute value of the test statistic?

Suppose the S&P 500 currently has a level of 875. The continuously compounded return on a 1-year T-bill is 4.25%. You wish to hedge an $800,000 stock portfolio that has a beta of 1.2 and a correlation of 1.0 with the S&P 500.

(a) What is the 1-year futures price for the S&P 500 assuming no dividends?

(b) How many S&P 500 futures contracts should you short to hedge your portfolio? What return do you expect on the hedged portfolio?

*YOU MUST ANSWER WITH DETAILED WORKING!!

Prior to eating breakfast, 25 participants were randomly assigned to eat a large meal, 25 to eat a small meal, and 25 to eat nothing. Immediately following the meal, all participants took a memory test. The means and estimated population variances for the three groups on the memory test were: Large: M = 2.1, S2 = 1.0; Small: M = 2.5, S2 = 1.2; Nothing: M = 2.9, S2 = 0.8. Using the .05 significance level, does amount of breakfast eaten affect memory? Perform a complete hypothesis test.

Assume analysts provide the following types of information. Further assume that short sales are allowed.

It is also known that the correlation coefficient between A and B is ρ=0.5.

a) What is the optimum portfolio of risky assets? What is the return and standard deviation of this optimal portfolio? [4.0]

b) Suppose you have $1000. How much should you invest in this portfolio and the risk-free asset if you wish to get 20% expected return? [1.0]