invest_3ed.pdf

# Serious we will fix the level of significance

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serious, we will fix the level of significance (controlling the probability of a Type I error), and then consider the probability of a Type II error. (p) What might the player ask for in order to have a better chance of showing that his success probability really has improved? Explain. (q) In the applet, change the sample size from 20 to 100 [keeping the number of samples at 1000] and press Draw Samples . How did the two distributions change?

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Chance/Rossman, 2015 ISCAM III Investigation 1.6 59 How did the overlap between the two distributions change? What does the change in amount of overlap mean in terms of how likely the player is to convince the manager that he has improved? (r) Use the applet to determine the rejection region corresponding to the 5% level of significance. In other words, how many hits would the 0.333 hitter need to get in 100 at-bats to convince the manger that he has improved (with a p-value < 0.05)? (s) What is the approximate power of the test for a 0.333 hitter with this cut-off value? How does this approximate power compare to what you found in (i) with a sample size of 20 at-bats? (t) So if the player really has improved, does increasing the sample size help the player? Explain why this makes intuitive sense. (u) Now determine the rejection region so that the probability of a Type I error (level of significance) is at most 0.01 (still with a sample size of 100 at-bats). What is the new cut-off value for the rejection region? What is the new empirical probability of a Type II error? Did this change help the player’s likelihood of convincing the manager that he has improved? Explain. (v) Now suppose the player actually became a 0.400 hitter, meaning that he increased his success probability to 0.4! How do you expect this to affect the power of the test (i.e., the player’s ability to convince the manager he has improved)? (w) Now specify 0.400 as the alternative value of ࠵? , the probability of success, (with n = 100), and press Draw Samples . How did the two distributions change? How did the overlap between the two distributions change? What is the rejection region so the probability of a Type I error is at most 0.05?
Chance/Rossman, 2015 ISCAM III Investigation 1.6 60 How does it compare to the first rejection region? What is the approximate power in this case? How does this probability compare to what you found in (s)? (x) Write a paragraph summarizing your findings. Include a discussion of how the trade-off between P(Type I error) and P(Type II error) relates to the trade-off between making the player happy and making the manager happy with the decision making process. [ Hint : What happens if we try to decrease the probability of one of the errors?] Also discuss how these errors are affected by changes in sample size, level of significance, and the alternative probability of success (the player’s true improved performance).

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