1. Identify the appropriate test statistic or statistics for conducting the following hypothesis tests. (Clearly identify the test statistic and, if applicable, the number of degrees of freedom. For example, We conduct the test using an x- statistic with y degrees of freedom.)
   1. H₀: m = 0 versus H_a: m 0, where m is the mean of a normally distributed population with unknown variance. The test is based on a sample of 15 observations.
   2. H₀: m = 0 versus H_a: m 0, where m is the mean of a normally distributed population with unknown variance. The test is based on a sample of 40 observations.
   3. H₀: m 0 versus H_a: m > 0, where m is the mean of a normally distributed population with known variance s2. The sample size is 45.
   4. Identify and describe which is error (Type I or Type II) is present in the following scenarios: 1) Adoctor declares that his patient, a man, is pregnant.

   5. H0: s2 = s2 versus H

2. Identify and describe which is error (Type I or Type II) is present in the following scenarios: 1) Adoctor declares that his patient, a man, is pregnant.

3. 2) A doctor declares his patient, an 8.5 month pregnant woman, is not pregnant. She then goes into labor when she is told the news.

   Saffa Tyres is a manufacturer in a mature cyclical industry. During the most recent industry cycle, its net income averaged $30 million per year with a standard deviation of $10 million (n = 6 observations). Management claims that Saffas performance during the most recent cycle results from new approaches and that we can dismiss profitability expectations based on its average or normalized earnings of $24 million per year in prior cycles.
   1. With m as the population value of mean annual net income, formulate null and alternative hypotheses consistent with testing Saffa managements claim.
   2. Assuming that Saffas net income is at least approximately normally distributed, identify the appropriate test statistic.
   3. If this were a two tailed Z-test, with 95% confidence what would be your rejection point or critical value?
   4. Assuming a critical value of 2.015, determine whether or not to reject the null hypothesis at the 0.05 significance level.

4. A speculator has identified 300 token coins to invest in. The speculator wishes to have aportfolio of 30 coins. 30% of the stock are identified as bit- coin based, 45% as etherium basedand 11% as other. Each individual coin has a 30% chance of increasing in value of the next year:
   1. How many different ways could the 3 types of coin be combined within the portfolioof 30 coins if the order theyre combined is not important?
   2. How many different ways could the 3 types of coin be combined within the portfolio of 30 coins if the order theyre combined is important?

3. Assuming a binomial distribution what is probability that exactly 12 of the 30 coins in the portfolio increase in value of the next year?

5. Use the following table for the next three questions:

Mean Return (%)		Standard Deviation of Returns (%)	Skewness	Excess Kurtosis
Portfolio 1	8	15	0.0	0.7
Portfolio 2	11	21	0.9	1.9
Portfolio 3	13	30	1.7	6.3

1. An investment adviser bases his allocation on the Sharpe ratio. Assuming a risk-free rate of 1.5%, which portfolio is he most likely to recommend?

2. The skewness of Portfolio 1 indicates its mean return is most likely less than, equal or greater than the median?

3. Compared with a normal distribution, the distribution of returns for Portfolio 3 most likely is less peaked, have a greater number of extreme returns or have fewer small deviations from its mean?

6. An investor is considering an annuity that pays $40,000 per year for four years.
   1. Assuming the first $40,000 is paid in a years time given a discount rate of 4% what should the investor pay for this annuity today?
   2. Assuming the first $40,000 is paid out immediately what should the investor pay today? (Asssume same discount rate).

3. If the investor pays $130,000 today and assuming the first payment arrives in a years time, what would this investments internal rate of return be? (Asssume same discountrate).
4. What would the investor pay today if the first payment arrived in 5 years time? (Asssume same discount rate).
5. If the investor invests $40,000 per year at the end of the next 4 years what would this be worth in 4 years time? (Asssume same discount rate).