Question

6. Assume that adults have IQ scores that are normally distributed with mean 100 and standard deviation 15. In each case, draw the graph (optional), then find the probability of the given scores. ROUND YOUR ANSWERS TO 4 DECIMAL PLACES

a. Find the probability of selecting a subject whose score is less than 115. __________

b. Find the probability of selecting a subject whose score is greater than 131.5. __________

c. Find the probability of selecting a subject whose score is between 90 & 110. __________

d. Find the probability of selecting a subject whose score is between 110 & 120. _________

e. Find the probability of selecting a subject whose score is greater than 105. ____________

Answer 1

a)

X ~ N ( µ = 100 , σ = 15 )
We convert this to standard normal as
P ( X < x ) = P ( Z < ( X - µ ) / σ )
P ( ( X < 115 ) = P ( Z < 115 - 100 ) / 15 )
= P ( Z < 1 )
P ( X < 115 ) = 0.8413

b)

P ( X > 131.5 ) = P(Z > (131.5 - 100 ) / 15 )
= P ( Z > 2.1 )
= 1 - P ( Z < 2.1 )
= 1 - 0.9821
= 0.0179

c)

P ( 90 < X < 110 ) = P ( Z < ( 110 - 100 ) / 15 ) - P ( Z < ( 90 - 100 ) / 15 )
= P ( Z < 0.67) - P ( Z < -0.67 )
= 0.7486 - 0.2514
= 0.4971

d)

P ( 110 < X < 120 ) = P ( Z < ( 120 - 100 ) / 15 ) - P ( Z < ( 110 - 100 ) / 15 )  
= P ( Z < 1.33) - P ( Z < 0.67 )   
= 0.9082 - 0.7486  
= 0.1597  

e)

P ( X > 105 ) = P(Z > (105 - 100 ) / 15 )
= P ( Z > 0.33 )
= 1 - P ( Z < 0.33 )
= 1 - 0.6293
= 0.3707