Question

Suppose you are taking a test in Math 1044, which is made up of 100 independent, identically distributed students. Your professor claims that the average score will be 72 with a standard deviation of 10 points. Let S be the normal approximation to the average score and use the Central Limit Theorem to answer the following questions. Please simplify answer in terms of the Φ function before entering them in a calculator.C

(a) How is S distributed and what is the CDF of S?

(b) Find the probability that the average score is below a 70.

(c) Find the probability that the actual average score is between 68 and 72.

(d) Find the probability that the average score is above a 74.

Answer 1

a)

S distributed normally.

CDF of S

P( <= x) = P(Z < ( x - ) / ( / sqrt(n) ) )

b)

Given, µ = 72 , σ = 10
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 70 - 72 ) / ( 10 / √100 )
Z = -2
P ( ( X - µ ) / ( σ/√(n)) = ( 70 - 72 ) / ( 10 / √(100) )
P ( < 70 ) = P ( Z < -2 )
= 0.0228

c)

P(68 < < 72) = ?
Standardizing the value   
Z = ( X - µ ) / ( σ / √(n))  
Z = ( 68 - 72 ) / ( 10 / √(100))  
Z = -4  
Z = ( 72 - 72 ) / ( 10 / √(100))  
Z = 0  
P ( 68 < X̅ < 72 ) = P ( Z < 0 ) - P ( Z < -4 )   
P ( 68 < X̅ < 72 ) = 0.5 - 0  
P ( 68 < X̅ < 72 ) = 0.5  

d)

P( > 74) = ?
Standardizing the value   
Z = ( X - µ ) / ( σ / √(n))  
Z = ( 74 - 72 ) / ( 10 / √ ( 100 ) )  
Z = 2  
P ( ( X - µ ) / ( σ / √ (n)) > ( 74 - 72 ) / ( 10 / √(100) )  
P ( Z > 2 )   
P ( X̅ > 74 ) = 1 - P ( Z < 2 )   
P ( X̅ > 74 ) = 1 - 0.9772  
P ( X̅ > 74 ) = 0.0228