Question

Candidates are applying for an elite college. From prior statistics we know that white students have an acceptance rate of 24% while Asian students have an acceptance rate of only 17%. A random sample of 100 White students and 100 Asian students were selected. Let X be the number of White applicants (out of the 100) who are accepted. Let Y be the number of Asian applicants (out of the 100) who are accepted.

Question 1 of 4 Which of the following is true regarding the random variables X and Y?

Neither X nor Y can be well-approximated by a normal random variable.

Only Y can be well-approximated by a normal random variable.

Both X and Y can be well-approximated by normal random variables.

Only X can be well-approximated by a normal random variable.

The remaining questions refer to the following information: Suppose the scores on an exam are normally distributed with a mean μ = 75 points, and standard deviation σ = 10 points.

Question 2 of 4 Suppose that the top 4% of the exams will be given an A+. In order to be given an A+, an exam must earn at least what score? 65 , 68 , 85 , 92.5

Question 3 of 4 What is the exam score for an exam whose z-score is 1.25? 60 , 70 , .1056 , 87.5 , .8944

Question 4 of 4 The instructor wanted to "pass" anyone who scored above 69. What proportion of exams will have passing scores? 0.6 , -0.6 , .7257 , .2743

Answer 1

1)

for normal approximation, np>5 , nq > 5

for x

np = 100 * .24 = 24 >5

nq = 100 * .66 =66 > 5

for y

np= 100*.17 = 17>5

nq = 100*.83 = 83> 5

so option (C ) is correct

Both X and Y can be well-approximated by normal random variables.

2)

µ=   75                  
σ =    10                  
proportion= 1-0.04 = 0.96                  
                      
Z value at    0.96   =   1.75   (excel formula =NORMSINV(   0.96   ) )
z=(x-µ)/σ                      
so, X=zσ+µ=   1.75   *   10   +   75  
X   =   92.51   (answer)          

3)

µ=   75
σ=   10
Z=(X-µ)/σ=   (X-75)/10=       1.25

X-75 = 12.5 +75 =  87.5

X= 87.5

4)

µ =    75                  
σ =    10                  
                      
P ( X ≥   69.00   ) = P( (X-µ)/σ ≥ (69-75) / 10)              
= P(Z ≥   -0.600   ) = P( Z <   0.600   ) = 0.7257   (answer)

SO
proportion of exams will have passing scores =   0.7257   (answer)