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.