A manufacturer makes ball bearings that have a target diameter of 5\text{ mm}5 mm5, start text, space, m, m, end text. Quality control engineers sample ball bearings from a batch to test the hypotheses H_0: \mu = 5\text{ mm}H 0 :μ=5 mmH, start subscript, 0, end subscript, colon, mu, equals, 5, start text, space, m, m, end text vs. H_\text{a}: \mu \neq 5\text{ mm}H a :μ =5 mmH, start subscript, start text, a, end text, end subscript, colon, mu, does not equal, 5, start text, space, m, m, end text, where \muμmu is the true mean diameter of the ball bearings in that batch. Suppose that H_0H 0 H, start subscript, 0, end subscript is actually true. Which situation below would have the lowest probability of a Type I error?

Afriend of mine is giving a dinner party. his current wine supply includes 10 bottles of zinfandel, 8 of merlot, and 11 of cabernet (he only drinks red wine), all from different wineries. (a) if he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this? ways (b) if 6 bottles of wine are to be randomly selected from the 29 for serving, how many ways are there to do this? ways (c) if 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety? ways (d) if 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? (round your answer to three decimal places.) (e) if 6 bottles are randomly selected, what is the probability that all of them are the same variety? (round your answer to three decimal places.)