On a planet far far away from Earth, IQ of the ruling species is normally distributed with a mean of 103 and a standard deviation of 18. Suppose one individual is randomly chosen.

Question

On a planet far far away from Earth, IQ of the ruling species is normally distributed with a mean of 103 and a standard deviation of 18. Suppose one individual is randomly chosen. Let X = IQ of an individual. 

a. What is the distribution of X? X ~ N(,)  

b. Find the probability that a randomly selected person's IQ is over 122.  Round your answer to 4 decimal places.

c. A school offers special services for all children in the bottom 2% for IQ scores. What is the highest IQ score a child can have and still receive special services? Round your answer to 2 decimal places. 

d. Find the Inter Quartile Range (IQR) for IQ scores. Round your answers to 2 decimal places.

Q1:   

Q3:   

IQR:   

Answer 1

a) X is normal with mean 103 and variance 18^2, so X~N(103, 324)

b) P(X > 122) = P( (x-µ)/σ > (122 - 103)/18 ) = P(z > 19/18) ≈ 0.1456

c) cutoff is z = -2.053749 by table or calculator; set z = (x-µ)/σ, so -2.053749 = (x - 103)/18 and therefore, by algebra, cutoff value for x is 66.03252, so the highest score with which a student will still be in the bottom 2% is 66.03 rounded to two decimals. Note that there is no such thing as fractional IQ scores.

d) using calculator figure the 75% (Q3) and 25% (Q1) points in this distribution; The IQR is (Q3 - Q1) = 115.1408 - 90.85918 = 24.28162 or 24.28, rounded as required