Critics of the U.K. lottery described in Exercise 9.51 claimed that the draws were not random. A supporter claimed that the draws were random.

a. If the critics were wrong, did they commit a Type I or Type II Error?

b. If the supporter was wrong, did she commit a Type I or Type II Error?

c. Which type of error would cause more inconvenience to the lottery's organizers, Type I or Type II? Explain.

"Study Proves Number Bias in UK Lottery" reported on the Internet about a document completed in 2002, "The Randomness of the National Lottery," which was meant to offer irrefutable proof that it was random. But the authors hit a snag when they found that one numbered ball, 38, was drawn so often that they believed there was bias in the selections, perhaps due to a physical difference in the ball: "Haigh and Goldie found that over 637 draws you would expect each number to be drawn between 70 and 86 times. But they found that 38 came up 107 times, 14 times more than the next most drawn ball. 'That some number has been drawn 107 times is very unusual-the chance is under 1%.'"30

a. If each number is expected to be drawn between 70 and 86 times, on average how many times is it expected to be drawn?

b. Based on your answer to part (a), what proportion of times is each number expected in 637 draws? (Round to the nearest thousandth.)

c. Show that standard deviation of sample proportion in this situation would be 0.013.

d. What proportion of the 637 draws did the number 38 come up? (Round to the nearest thousandth.)

e. Find z, the standardized difference between sample proportion (your answer to part [d]) and ideal proportion (your answer to part [b]).

f. Were the authors correct that the chance is under 1%?

g. A spokesperson defending the lottery's randomness criticized the authors' conclusions: "If you trawl through an awful lot of data you are always going to uncover patterns somewhere." Is her statement consistent with the decisionmaking process of hypothesis testing? Explain.