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# P1.T2.314. Miller's one- and two-tailed hypotheses

#### Nicole Seaman

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AIMs: Define and interpret the null hypothesis and the alternative hypothesis, and calculate the test statistics. Define, interpret, and calculate the t-statistic. Differentiate between a one-tailed and a two-tailed test and explain the circumstances in which to use each test.

Questions:

314.1. You are given the following sample of annual returns for a portfolio manager: -6.0%, -3.0%, -2.0%, 0.0%, 1.0%, 2.0%, 4.0%, 5.0%, 7.0%, 10.0%. The sample mean of these ten (n = 10) returns is +1.80%. The sample standard deviation is 4.850%. The sample mean is positive, but how confident are we that the population mean is positive? (note: this is a simplified version of Miller's problem 5.2, since it provides the sample mean and standard deviation, but it nevertheless does require calculations/lookup)

a. t-stat of 1.17 implies one-sided confidence of about 86.5%
b. t-stat of 1.29 implies two-sided confidence of about 88.3%
c. t-stat of 2.43 implies one-sided confidence of about 90.7%
d. t-stat of 3.08 implies two-sided confidence of about 97.4%

314.2. A sample of 25 money market funds shows an average return of 3.0% with standard deviation also of 3.0%. Your colleague Peter conducted a significance test of the following alternative hypothesis: the true (population) average return of such funds is GREATER THAN the risk-free rate (Rf). He concludes that he can reject the null hypothesis with a confidence of 83.64%; i.e., there is a 16.36% chance (p value) that the true return is less than or equal to the risk-free rate. What is the risk-free rate, Rf? (note: this requires lookup-calculation)

a. 1.00%
b. 1.90%
c. 2.00%
d. 2.40%

314.3. A random sample of twenty (n = 20) publicly-traded retailers produces a sample average price-to-earnings (P/E) ratio of 20.00 with sample standard deviation of 8.50. We are interested in testing hypothesis related to a possible population mean of 15.0. Each of the following is a valid conclusion EXCEPT which is not?

a. With 95.0% confidence, we reject a two-sided null hypothesis that the population's mean P/E ratio is 15.0
b. With 99.0% confidence, we reject a two-sided null hypothesis that the population's mean P/E ratio is 15.0
c. With 95.0% confidence, we accept a one-sided alternative hypothesis that the population's mean P/E ratio is greater than 15.0
d. With 99.0% confidence, we accept a one-sided alternative hypothesis that the population's mean P/E ratio is greater than 15.0