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1. Most manufacturing processes create defective items. Often these are tolerated up to a certain point, after which machines must be replaced, a costly and time-consuming process. Suppose you are working at a company that will permit 5% of all items to be defective. Your boss is curious if things have gotten worse and asks you to inspect 230 random items.
a. If you find 16 items are defective, what advice should you give your boss based on a hypothesis test with a = 0.04? As always, follow the steps from class.
b. In R, you can conduct the test quite easily with the prop.test command. Read the documentation for this and write a single line of code that reproduces your results rom part a. (Note: Set the "continuity correction" to false.)
c. Your boss is wondering if an exact test for this situation would give different results. Find the P-value based on an exact test without using binom.test and then with binom.test in R (you'll get the same answer).
2. Does the idea known as "home-field advantage" (HFA) actually exist? HFA suggests that in a given sport, the home team will beat the away team more often than half the time. Several theories have been offered for why this might occur: familiarity with your arena/playing space, support of the home crowd, refereeing that favors the home team, etc. To explore HFA, researchers looked at 1000 random NFL games in the last 40 years and found that in 573 cases, the home team won.
a. Draw a conclusion about the idea of HFA using an approximate test with a = 0.01, and show that you have met the conditions necessary for using this test.
b. After publication of the findings from part a, you read on an NFL blog that "Data show the existence of HFA, likely the result of biased refereeing." Respond to this claim from a statistical perspective.
3. People often look down on machine learning because in some settings, it can only improve things in small increments. While this might be true, in many settings a slight change can have a huge impact. As an example, Americans spend about 3 trillion dollars per year spread across 30 billion credit card transactions. Suppose that 0.5% (half a percent) of these transactions are fraudulent, and hence, credit card companies lose money reimbursing their users. If machine learning could help reduce the percentage of fraudulent claims even slightly, this would save companies billions of dollars! Researchers at Visa have designed a new algorithm to predict fraud and are curious if it has reduced illegal card usage. You are tasked with determining whether this claim is statistically reasonable using an approximate test with a = 0.05.
a. What is the smallest number of transactions you could look at and meet Larsen&Marx's requirements for using the approximate test?
b. Suppose you end up looking at 2019 claims. What is the largest number of fraudulent claims that could appear among those 2019 claims but you would still move to the alternative hypothesis?
4. What hypothesis test has duality with the CI: (-xo, X+ 1.50x)? Assume X ~ N(H,02) with o known.
5. Is it possible that a = B in a hypothesis test? Suppose we have X ~ N(, 42) with H0: H = 10 and H1: u > 10. If you collect a sample of size 16, for what a will B = a, assuming the true value of H is 12? Include a beautiful picture in your answer.
6. Suppose you are studying a phenomenon that is well-modeled by N (, 32). . Existing research claims that H = 20, but you think u might be lower based on recent changes in society. You'd like to collect some data to verify your claim and plan to use a sample of size 60. If u is actually 19, what should you set a to in your hypothesis test if you want a Type II error rate of 0.08?
Include a beautiful picture in your answer.

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