Question

Suppose the probability that Joe Schmo makes a single free throw is .37, and suppose he shoots n = 12 free throws in a particular game. Let Y denote the number of made free throws.

a. The distribution of Y is:

A. skewed negatively B. not skewed C. skewed positively

b. Use the binomial formula to compute the probability that Joe Schmo will make exactly 3 free throws. (5 decimals)

     

c. What are the least and most likely number of free throws made by Joe Schmo?

d. What is the exact binomial probability that Joe Schmo makes at least half of his free throws? (4 decimals)

e. Show that the mean and variance of Y are 4.44 and 2.7972, respectively.

f. What is the normal approximation (with continuity correction) to the exact probability of Part 2 above? (5 decimals)

g. Determine whether the “guidelines” for assessing whether the sample size is “sufficiently large” for a “good normal approximation” are satisfied or not.

Answer 1

(Since there are more than 4 parts i will answer first 4 parts)

probability distribution in binomial :

the binomial plot for p =0.37 and n=12 is :

a.

we can see probability is more in right side so it is positively skewed

b.

P(3) = 12C3 * (0.37^3) * (1-0.37)^(12-3)

P(3) = 0.1742

c.

as we can see in the plot of probability :

least likely = 12

most likely = 4

d.

P(atleast half) = P(x>=6) = P(6)+P(7)+P(8)+.....+P(12)

P(atleast half) = 0.2588

P.S. (please upvote if you find the answer satisfactory)