Question

The percentage of students with a GPA of 3.0 or higher is 15%. Suppose the random variable X represents the total number of students with a GPA of 3.0 or higher in a random sample of 500 students.

a) Find the mean of X. (round to the nearest whole number)

b) Find the standard deviation of X. (round to the nearest whole number)

c) Determine the shape of the distribution for X. (letter only)

A) skewed-left, since p > .5 and n is small.

B) skewed-right, since p < .5 and n is small.

C) bell-shaped, since p = .5 and n is small.

D) normal, since n ⋅ p ⋅ ( 1 − p ) ≥ 10.

d) Based on your answers from parts (a)-(c), would it be unusual for 80 students in the sample to have a GPA of 3.0 or higher. (letter only)

A) Yes, since P ( 80 ) < .05

B) Yes, since 80 is more than 2 standard deviations away from the mean

C) No, since 80 is within 2 standard deviations of the mean

D) No, since anyone can get a 3.0 or higher.

E) No, since P ( 80 ) ≥ .05.

e) Based on your answers from parts (a)-(c), would it be unusual for 95 students in the sample to have a GPA of 3.0 or higher. (letter only)

A) Yes, since P ( 95 ) < 0.05.

B) Yes, since 95 is more than 2 standard deviations away from the mean.

C) No, since 95 is within 2 standard deviations of the mean.

D) No, since anyone can get a 3.0 or higher.

E) No, since P ( 95 ) ≥ .05.

Answer 1

Solution:
Given in the question
P(Students with a GPA of 3.0 or higher) p= 0.15
Number of sample p= 500
Solution(a)
The mean of X can be calculated as np = 0.15*500 = 75
Solution(b)
Standard deviation of X can be calculated as
sqrt(np(1-p)) = sqrt(500*0.15*(1-0.15)) = sqrt((500*0.15*0.85) = 8
Solution(c)
Our distribution is approximately normal because np*(1-p) = 500*0.15*(1-0.15) = 64 which is greater than 10. So its answer is D.
Solution(D)
We need to calculate would it be unusual for 80 students in the sample to have a GPA of 3.0 or higher?
Here we will use the standard normal distribution as follows
Z = (p^-p)/sqrt(p*(1-p)/n)) = ((80/500)-0.15)/sqrt(0.15*(1-0.15)/500)) = 0.626
So its correct answer is C.i.e. No since 80 is within 2 standard deviations of the mean.
Solution(E)
We need to calculate would it be unusual for 95 students in the sample to have a GPA of 3.0 or higher?
Here we will use the standard normal distribution as follows
Z = (p^-p)/sqrt(p*(1-p)/n)) = ((95/500)-0.15)/sqrt(0.15*(1-0.15)/500)) = 2.50
So its answer is B i.e. Yes since 95 is more than 2 standard deviations away from the mean.