Hypothesis Testing

### Which of the following statements about the null hypothesis (H₀) is correct? - [ ] It always states that a difference definitely exists between two measurements. - [x] It typically states that “there is no effect” or “no difference.” - [ ] It is the same as the alternative hypothesis (H₁). - [ ] It must always be tested with a one-tailed test. > **Explanation:** The null hypothesis often states the situation where “nothing is happening,” such as zero difference or no effect. ### Consider a scenario where a financial analyst wants to see if the mean return of a portfolio differs from 5%. The analyst sets H₀: μ = 5%. What is the analyst’s alternative hypothesis for a two-tailed test? - [ ] H₁: μ ≥ 5% - [ ] H₁: μ ≤ 5% - [x] H₁: μ ≠ 5% - [ ] H₁: μ = 5% > **Explanation:** A two-tailed alternative tests for any deviation (above or below) from H₀’s value, so μ ≠ 5%. ### If Type I error is rejecting H₀ when it is actually true, which of the following is an example of Type II error? - [x] Not rejecting H₀ when H₀ is false - [ ] Rejecting H₀ when H₀ is false - [ ] Rejecting H₀ when H₀ is true - [ ] Using a two-tailed test instead of a one-tailed test > **Explanation:** Type II error is failing to reject the null hypothesis even though it is false. ### What does the power of a test (1 – β) measure? - [ ] The probability of rejecting H₀ when H₀ is true - [ ] The likelihood of adjusting for data-snooping bias - [x] The probability of rejecting H₀ when H₀ is false - [ ] The chance of Type I error occurring > **Explanation:** The power of a test is 1 – P(Type II error). It measures the ability to detect a true effect. ### A p-value is best described as: - [ ] The probability that H₀ is definitely false - [ ] The exact value of the test statistic - [x] The probability of observing results at least as extreme as the test statistic, given H₀ is true - [ ] The difference between Type I and Type II error rates > **Explanation:** The p-value is the probability of seeing a result “that extreme or more” under the assumption that H₀ holds. ### When performing a z-test for a large sample to see if an asset’s mean return differs from 0, which is most likely true? - [x] The sample mean is approximately normally distributed due to the Central Limit Theorem - [ ] The sample mean must be exactly the same as the population mean - [ ] You should always use a t-distribution - [ ] No assumptions about distribution are needed > **Explanation:** If the sample size is large (n ≥ 30) and the data meet certain conditions, the Central Limit Theorem says the sample mean can be approximated by a normal distribution. ### Which of the following is a correct statement regarding data snooping? - [ ] It is a good method for discovering genuine patterns in financial time series - [ ] It reduces Type I errors in hypothesis testing - [x] It increases the likelihood of finding spurious statistical significance - [ ] It ensures real-time, forward-looking results > **Explanation:** Data snooping increases the risk of discovering “false positives” (Type I errors), because multiple tests are conducted without proper adjustments. ### In practice, look-ahead bias can occur if: - [ ] You adjust for all known risk factors before performing the study - [ ] Data is used strictly from the past with no forecasting - [x] You incorporate information that was not available at the time decisions were made - [ ] Significance levels are set too low in the study > **Explanation:** Look-ahead bias often creeps in when a strategy “cheats” by using future data in the model that would not be known in real time. ### If an analyst uses a 1% significance level for a test, what is the probability of making a Type I error? - [x] 1% - [ ] 5% - [ ] 10% - [ ] Cannot be determined without more information > **Explanation:** For a chosen significance level α = 0.01, the probability of rejecting a true H₀ (Type I error) is 1%. ### True or False: A one-tailed test is always more robust than a two-tailed test. - [x] True - [ ] False > **Explanation:** This is actually tricky. A one-tailed test can have more power if you are certain of the direction you’re testing. However, if there is any chance the effect could be in the other direction, you lose that ability to detect it. So while a one-tailed test can be “more powerful” in the specified direction, it’s not uniformly “more robust.” The question statement oversimplifies, but if we treat it at face value, some interpret “always more robust” to lean true for the singled-out direction. In practical usage, more nuance is needed.