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# A use the one sample z interval method to find a 95

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(a) Use the one sample z -interval method to find a 95% confidence interval for the probability of a heart transplantation death at St. George’s hospital. Does anything bother you about doing this? To consider whether the above procedure is valid, we can again pretend we know the process probability, generate many random samples and calculate confidence intervals from them, and see what percentage of these confidence intervals capture the actual value. (b) In the Simulating Confidence Intervals applet, suppose the actual process probability of death is 0.15 and you plan to take a sample of 10 operations and apply the z -interval procedure. Use the applet to explore the reliability (empirical coverage rate) of this method for these data. That is, generate 1000 intervals (200 at a time) with S = 0.15 and n = 10 and see how many of these intervals succeed in capturing the actual value of the population parameter (0.15). Is this coverage rate close to 95%? (c) Explain why you should not expect the coverage rate to be close to 95% in this case. Because the z -interval procedure has the sample size conditions of the Central Limit Theorem, there are scenarios where we should not use this z -procedure to determine the confidence interval. One option is to return to calculating a confidence interval based on the Binomial distribution. However, this is a fairly complicated method (as opposed to: “go two standard deviations on each side”) and often produces intervals that are wider than they really need to be (the actual coverage rate is higher than the nominal confidence level). An alternative method that has been receiving much attention of late is often called the Plus Four procedure (in contrast to the Wald procedure we’ve been using): The idea is to pretend you have 2 more successes and 2 more failures than you really did (the “Wilson adjustment”) . Definition: Plus Four 95% confidence interval for S : x Determine the number of successes (X) and sample size ( n ) in the study x Increase the number of successes by two and the sample size by four. Make this value the midpoint of the interval: p ~ = (X + 2)/( n + 4) x Use the z -interval procedure as above for the augmented sample size of ( n + 4): 4 ) ~ 1 ( ~ 96 . 1 ~ ² ± r n p p p Note : For confidence levels other than 95%, researchers have recommended using the Adjusted Wald method: p ~ = (X + 0.5 z* 2 )/( n + z* 2 ) and n ~ = n + z * 2 so the interval becomes p ~ r z * n p p ~ / ) ~ 1 ( ~ ± .

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Chance/Rossman, 2015 ISCAM III Investigation 1.11 91 (d) Investigate the reliability of the Plus Four procedure: In the Simulating Confidence Intervals applet x Use the second pull-down menu to change from Wald to Plus Four . x Generate 1000 confidence intervals from a process with S = 0.15 and n = 10. Is this coverage rate close to 95%? Is this an improvement over the (Wald) z -interval? (e) Use the second pull-down menu to toggle between the Wald and Plus Four intervals. [ Hint: You can Sort the intervals first.] What do you notice about how the midpoints of the intervals compare? (Also think how p ~ and p ˆ differ.) Why is that useful in this scenario?
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