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# P1.T2.719. One- versus two-tailed hypothesis tests (Miller Ch.7)

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Learning objectives: Differentiate between a one-tailed and a two-tailed test and identify when to use each test. Interpret the results of hypothesis tests with a specific level of confidence. Demonstrate the process of backtesting VaR by calculating the number of exceedances.

Questions:

719.1. Barbara observes a sample with the following statistics: mean of X and standard deviation of Y. She thinks the true (aka, population) mean is Z. She does NOT know the population's variance nor does she even know the population's distribution, including she cannot assume it is normal. If she wants to conduct a hypothesis test of the observed sample mean, each of the following statements is true EXCEPT which is incorrect?

a. If the sample is small (ie, n < 30), there is no valid test statistic
b. If the sample is large, she is justified in conducting a hypothesis test of the mean with a student's t distribution (and she may even approximate the student's t with a normal distribution)
c. A one-tailed test is more appropriate if (i) the sample mean is either greater than, or less than, the null hypothesized value or (ii) the metric is financial such as in this case of a mutual fund's mean return
d. If the sample is large, there is a valid test statistic and for a given observed X either (i) a switch from two-tailed to one-sided confidence and/or (ii) an increase in the sample size will increase the likelihood of rejecting the null

719.2. MallRat is a real estate investment trust (REIT). During calendar year 2017, MallRat produced an average weekly (n = 52) return of +20 basis points with volatility of 19 basis points. Its benchmark during the same period produced an average return of +15 basis points. With 95.0% confidence, is the difference statistically significant?

a. The result is different than the benchmark regardless of the test: the null is rejected in either a one-sided or a two-sided test
b. The result is NOT different than the benchmark (accept null in two-sided test), but it is greater than the benchmark (reject null in one-sided test)
c. The result is different than the benchmark (reject null in two-sided test), but it is NOT greater than the benchmark (accept null in one-sided test)
d. The result is NOT different than the benchmark regardless of the test: the null is accepted in either a one- and two-sided test

719.3. Sally the Risk Analyst conducted a backtest of her firm's value at risk (VaR) model. Her sample window included 120 trading days. In regard to her VaR backtest over 120 days, each of the following statements is true EXCEPT which is false?

a. If she uses the normal distribution to approximate the binomial, then six observed exceptions returns a test statistics of about 4.40
b. If she observed zero exceptions, at any realistic confidence level she will not be able to dismiss a 99.0% confident VaR model as bad; i.e., she cannot reject null hypothesis that 99.0% VaR model is good
c. If she observed three exceptions, at any realistic confidence level she will not be able to dismiss a 99.0% confident VaR model as bad; i.e., she cannot reject null hypothesis that 99.0% VaR model is good
d. Because the binomial distribution (implied by the backtest) is non-negative, a one-sided test is appropriate; if she prefers a two-sided test, then she should instead employ the negative binomial distribution