Question

1. What is the probability of selecting each of the following at random from the population of IQ scores (mean=100, SD=15)

1. A person whose IQ is between 90 and 120?

2. A person whose IQ is below 75?

3. A person with an IQ above 80 AND a person with an IQ below 110? (Note: you are looking for two people here -- one with an IQ above 80 and the other with an IQ below 110.)

Answer 1

Part A)

Values are given as,
Mean = 100,
Standard Deviation = 15,
x1 = 90,
x2 = 120,

We need to find P(90 < x < 120)
Using below formula


Put the values in the formula,
P[(90 - 100)/15 < z < (120 - 100)/15]
= (-0.67 < z < 1.33)
z scores are (-0.67, 1.33)

Using below formula,
P(a < z < b) = P(z < b) - P(z < a)

Put the values in above formula,
P(-0.67 < z < 1.33) = P(z < 1.33) - P(z < -0.67)

Using z table, we get,
P(z < 1.33) = 0.9082
P(z < -0.67) = 0.2514
So, P(-0.67 < z < 1.33) = 0.9082- 0.2514= 0.6568

Hence, P(-0.67 < z < 1.33) = 0.6568

Part B)

Values are given as,
Mean= 100,
Standard Deviation = 15,
x = 75,
We need to find P(x < 75)

Using below formula

Put the values in the formula,
P[(z < (75 - 100)/15]
= (z < -1.67)
z score is -1.67

Using z table, we get,
P(z < -1.67) = 0.0475

Hence, P(z < -1.67) = 0.0475

Part C)

Values are given as,
Mean= 100,
Standard Deviation = 15,
x1 = 80,
x2 = 110,

We need to find P(80 < x < 110)

Using below formula

Put the values in the formula,
P[(80 - 100)/15 < z < (110 - 100)/15]
= (-1.33 < z < 0.67)
z scores are (-1.33, 0.67)

Using below formula,
P(a < z < b) = P(z < b) - P(z < a)

Put the values in above formula,
P(-1.33 < z < 0.67) = P(z < 0.67) - P(z < -1.33)

Using z table, we get,
P(z < 0.67) = 0.74857
P(z < -1.33) = 0.09176
So, P(-1.33 < z < 0.67) = 0.7486- 0.0918= 0.6568

Hence, P(-1.33 < z < 0.67) = 0.6568