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# D if abe were to use a significance level of 005 and

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(d) If Abe were to use a significance level of 0.05 and Bianca were to use a significance level of 0.01, who would have a smaller probability of Type II error? Explain briefly.

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Chance/Rossman, 2015 ISCAM III Exploration 62 Probability Exploration: Exact Binomial Power Calculations We can use technology to obtain the rejection region and binomial probabilities of Type I and Type II errors based on the binomial distribution. The calculation of power will be a two-step process. x Step one: Determine the rejection region corresponding to the null hypothesis hypothesized value, the direction of the alternative hypothesis, and the level of significance. x Step two : Determine the probability of obtaining an observation in the rejection region for a specific alternative value of the parameter. To determine the rejection region we work “in reverse”: specifying a probability and asking the software to determine the corresponding number of successes. Technology Detour Determine the Rejection Region (Binomial) In R x The iscaminvbinom function (of the ISCAM workspace) takes the following input x alpha = the probability of interest (the level of significance) x n = the sample size (number of trials) x prob = the process probability ( S ) of the binomial distribution x lower.tail = TRUE or FALSE For example: > iscaminvbinom(alpha=.05, n=20, prob=.25, lower.tail=FALSE) should reveal a graph with the upper 5% of the distribution shaded in red, and X =… for the smallest cut-off value that has at most 0.05 probability above it (the actual probability above that cut-off value will also be displayed). In Minitab x Select Graph > Probability Distribution Plot x Select View Probability and press OK . x Use the pull-down menu to select the binomial distribution, specifying the values for the number of trials ( n = 20) and the hypothesized probability of success ( S = 0.250) for the inputs. x Press the Shaded Area tab and keep the Shaded Area defined by Probability . x Select the Right Tail graph and specify 0.05 as the probability value. x Press OK . (a) Based on the output from the technology, what is the smallest number of successes k such that P(X > k ) < 0.05 when n = 20 and S = 0.25? How does this compare to the simulation results? What is the exact probability of a Type I error in this case?
Chance/Rossman, 2015 ISCAM III Exploration 63 Now that we have the rejection region, we switch to the alternative probability of success and see how often we land in that rejection region. Technology Detour Calculating Power from Rejection Region (Binomial) In R x Use iscambinomprob as before, specify 0.333 as the probability of success, but staying in upper tail to match our alternative hypothesis. > iscambinomprob(k=9, n=20, prob=.333, lower.tail=FALSE) In Minitab x Select Graph > Probability Distribution Plot (View Probability) and use the pull-down menu to select the binomial distribution, specifying the values for the number of trials ( n = 20) and the hypothesized probability of success ( S = 0.333) for the inputs.

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