STA-290 HW 5

Link to IMS Textbook Chapter 14

Question 1: Suppose that before a study even begins, we decide that we will reject the null hypothesis, regardless of the evidence. What kind of error are we at risk of committing? What if instead we chose in advance to never reject the null hypothesis?

Question 2: What are the components/terms that make up the \(t\)-statistic? Explain what impact changing each of them would have on the test statistic.

Question 3: Suppose that we have have completed a study investigating the difference in average shell width of male and female horseshoe crabs where the computed \(t\)-statistic is \(t = 1.6\). Is this result statistically significant at a significance level of \(\alpha = 0.05\)? What additional information would you need to know in order to answer this conclusively. Explain.

Question 4: If I were to collect 100 samples and compute 100 95% confidence intervals, should I expect the length of each of them to be the same? What if I had bootstrapped the 100 confidence intervals instead? Justify your reasoning for each.

Question 5 (IMS #1): A patient named Diana was diagnosed with Fibromyalgia, a long-term syndrome of body pain, and was prescribed anti-depressants. Being the skeptic that she is, Diana didn’t initially believe that anti-depressants would help her symptoms. However, after a couple months of being on the medication she decides that the anti-depressants are working, because she feels like her symptoms are in fact getting better.

• Part A: Write the null hypothesis in words for Diana’s position when she started taking the medication
• Part B: What would be an example of a Type I error in this context?
• Part C: What would be an example of a Type II error in this context?

Question 6 (IMS #2) In each part below, there is a value of interest and two scenarios denoted (i) and (ii). For each part, report if the value of interest in larger under scenario (i), scenario (ii), or if they would be equal

• Part A: The standard error of \(\hat{p}\) (proportion) when (i) \(n = 125\) or (ii) \(n = 500\)
• Part B: The margin of error of a confidence interval when the confidence level is (i) 90% or (ii) 80 %
• Part C: The p-value for a z statistic of 2.5 calculated based on a sample of (i) \(n = 500\) or (ii) \(n = 1000\)
• Part D: The probability of making a Type II error when the alternative hypothesis is true and the significance level (\(\alpha\)) is (i) 0.05 or (ii) 0.10.