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In sports betting, Las Vegas sports books establish winning margins for a team that is favored to win a game. An individual can place a wager on the game and will win if the team bet upon wins after accounting for the spread. For example, if Team \(\mathrm{A}\) is favored by 5 points, and wins the game by 7 points, then a bet on Team \(A\) is a winning bet. However, if Team A wins the game by only 3 points, then a bet on Team \(A\) is a losing bet. In games where a team is favored by 12 or fewer points, the margin of victory for the favored team relative to the spread is approximately normally distributed with a mean of 0 points and a standard deviation of 10.9 points. Source: Justin Wolfers, "Point Shaving: Corruption in NCAA Basketball" (a) Explain the meaning of "the margin of victory relative to the spread has a mean of 0 points." Does this imply that the spreads are accurate for games in which a team is favored by 12 or fewer points? (b) In games where a team is favored by 12 or fewer points, what is the probability that the favored team wins by 5 or more points relative to the spread? (c) In games where a team is favored by 12 or fewer points, what is the probability that the favored team loses by 2 or more points relative to the spread?

In games where a team is favored by more than 12 points, the margin of victory for the favored team relative to the spread is normally distributed with a mean of -1.0 point and a standard deviation of 10.9 points. Source: Justin Wolfers, "Point Shaving: Corruption in NCAA Basketball' (a) In games where a team is favored by more than 12 points, what is the probability that the favored team wins by 5 or more points relative to the spread? (b) In games where a team is favored by more than 12 points, what is the probability that the favored team loses by 2 or more points relative to the spread? (c) In games where a team is favored by more than 12 points, what is the probability that the favored team "beats the spread"? Does this imply that the possible point shaving spreads are accurate for games in which a team is favored by more than 12 points?

The number of chocolate chips in an 18 -ounce bag of Chips Ahoy! chocolate chip cookies is approximately normally distributed, with a mean of 1262 chips and a standard deviation of 118 chips, according to a study by cadets of the U.S. Air Force Academy. Source: Brad Warner and Jim Rutledge, Chance \(12(1): 10-14,1999\) (a) Determine the 30 th percentile for the number of chocolate chips in an 18 -ounce bag of Chips Ahoy! cookies. (b) Determine the number of chocolate chips in a bag of Chips Ahoy! that make up the middle \(99 \%\) of bags. (c) What is the interquartile range of the number of chips in Chips Ahoy! cookies?

The ACT and SAT are two college entrance exams. The composite score on the ACT is approximately normally distributed with mean 21.1 and standard deviation 5.1 . The composite score on the SAT is approximately normally distributed with mean 1026 and standard deviation \(210 .\) Suppose you scored 26 on the \(\mathrm{ACT}\) and 1240 on the SAT. Which exam did you score better on? Justify your reasoning using the normal model.