AP Statistics: Homework & Solutions – Section 7.2 [Means and Variances of Random Variables] 7.24: Grade Distribution Here is a distribution of grades [A = 4, B = 3, and so on] for a large class of college students. Grade Probability 0 0.01 1 0.05 2 0.30 3 0.43 4 0.21 Find the average grade in the course [mean]. SOLUTION - 7.24: Grade Distribution Remember, sampling distributions of statistics are the probability distributions of RV [random variables]. The table gives a probability distribution for the discrete random variable for grades [with A = 4, B = 3, and so on]. To find the “average” is to find the weighted mean or expected value for the long run population. Mean µ = Σ 𝑥𝑖 ∗ 𝑝𝑖 = (0 x .01) + (1 x .05) + (2 x .30) + (3 x .43) + (4 x .21) = 2.78 7.34: Overdue for a Hit Retired baseball player Tony Gwynn got a hit about 35% of the time over an entire season [.350 batting average!]. After he failed to hit safely in 6 straight at-bats, a TV commentator said, “Tony is due for a hit by the law of averages.” Is that right? Why? SOLUTION - 7.34: Overdue for a Hit The TV commentator is not correct. They make an incorrect application of the law of large numbers to a small number of at-bats for Tony Gwynn. Remember the fallacy! The law of small numbers is not legitimate – short sequences of random events do not show the same average behavior that appears in the long run.