Question

A) An Olympic archer is able to hit the bull’s eye 80% of the time. Assume each shot is independent of the others. She will shoot 6 arrows. Let X denote the number of bull’s eyes she makes.

Find the mean of the probability distribution of X. Do not round

B) The GPA of students at a college has a mean of 2.9 and a standard deviation of 0.3. Scores are approximately normally distributed.

Suppose that the top 6% of students are eligible for the Honors Program. Find the GPA which is the cutoff score for students to qualify for this program. Round to the nearest hundredth.

Answer 1

Solution(A)
Given in the question
P(Able to hit the bull's eye) = 80% of the time or 0.8
q(Not able to hit the bull's eye) = 1-0.8 = 0.2
Also, each shot is independent of the others so we will use the binomial probability distribution
No. of sample n = 6
So Mean of the probability distribution can be calculated as
Mean = No. of sample*P(Able to hit the bull's eye) = 0.8*6 = 4.8

Solution(B)
Given in the question GPA of students scores are normally distributed with
Mean() = 2.9
Standard deviation() = 0.3
Also given that top 6% of students are eligible for the honors programs so p-value = 0.94
From Z table we found Z-score = 1.55477
So cutoff score for students to qualify for the program is
X = + Z-score* = 2.9 + 1.55477*0.3 = 2.9 + 0.47 = 3.37
So the cutoff score is 3.37 for students to qualify for this program.