Question

Suppose IQ scores are approximately normally distributed with a mean of 100 and a standard deviation of 15. Answer the following questions.

Question 1

What percent of the population has an IQ that is above average?

Question 2

What percent of the population has an IQ below 110?

What is the calculated z-score?

What is the percentage?

Question 3

What percent of the individuals in the population have an IQ above 130? (Individuals in this category are sometimes classified as "gifted".)

What is the calculated z-score?

What is the percentage?

Question 4

If an individual is randomly selected from the population, then what is the probability that his/her IQ is between 110 and 130? (Your answer should be left as a decimal expressed to four decimal places. No percents!)

  

Answer 1

Solution :

Given that ,

mean = = 100

standard deviation = = 15

1)

P(x > 100) = 1 - P(x < 100)

= 1 - P((x - ) / < (100 - 100) / 15)

= 1 - P(z < 0)

= 1 - 0.5

= 0.5

Answer = 50%

2)

P(x < 110) = P((x - ) / < (110 - 100) / 15)

= P(z < 0.67)

= 0.7486

z score = 0.67

Answer = 74.86%

3)

P(x > 130) = 1 - P(x < 130)

= 1 - P((x - ) / < (130 - 100) / 15)

= 1 - P(z < 2)

= 1 - 0.9772   

= 0.0228

z score = 2

Answer = 2.28%

4)

P(110 < x < 130) = P((110 - 100)/ 15) < (x - ) /  < (130 - 100) / 15) )

= P(0.67 < z < 2)

= P(z < 2) - P(z < 0.67)

= 0.9772 - 0.7486

= 0.2286

Probability = 0.2286