What percentage of people has an IQ score between 40 and 160? (b) What percentage of people has an IQ score less than 80 or greater than 120? (c) What percentage of people has an IQ score greater than 140? (a) nothing% (Type an integer or a decimal.)

a) P ( 40 < X < 160 ) = 0.997

b) P ( 80 < X < 120 ) = 0.68

c) P ( X > 140 ) = 0.025

Step-by-step explanation:

Given:

- Mean of the sample u = 100

- Standard deviation of the sample s.d = 20

Find:

a) What percentage of people has an IQ score between 40 and 160?

b) What percentage of people has an IQ score less than 80 or greater than 120?

c)  What percentage of people has an IQ score greater than 140?

Solution:

- Declaring a random variable X is the IQ score from a sample of students.

Where, Random variable X follows a normal distribution as follows:

                                  X ~ N ( 100 , 20 )

- We will use the 68-95-99.7 Empirical rule that states:

                                 P ( u - s.d < X < u + s.d ) = 0.68

                                 P ( u - 2*s.d < X < u + 2*s.d ) = 0.95

                                 P ( u - 3*s.d < X < u + 3*s.d ) = 0.997

part a)

-The P ( 40 < X < 160 ) is equivalent to P (u - 3*s.d < X < u + 3*s.d ), as given by the Empirical rule stated above. The limits can be calculated to verify:

                            u - 3*s.d = 100 - 3* 20 = 40

                            u + 3*s.d = 100 + 3* 20 = 160

-Hence, from empirical rule we have P ( 40 < X < 160 ) = 0.997

part b)

- The P ( 80 < X < 120 ) is equivalent to P (u - s.d < X < u + s.d ), as given by the Empirical rule stated above. The limits can be calculated to verify:

                            u - s.d = 100 - 20 = 80

                            u + s.d = 100 + 20 = 120

-Hence, from empirical rule we have P ( 80 < X < 120 ) = 0.68

part c)

- The P ( X > 140 ) is can be calculated from P (u - 2*s.d < X < u + 2*s.d ), as given by the Empirical rule stated above. The limits can be calculated to verify:

                            u - s.d = 100 - 2*20 = 60

                            u + s.d = 100 + 2*20 = 140

- We know that the probability between the two limits is P ( 60 < X < 140 ) = 0.95. Also the remaining the probability is = 1 - 0.95 = 0.05. The rest of remaining probability is divided between two section of the bell curve.

                            P ( X < 60 ) = 0.025

                            P ( X > 140 ) = 0.025

- We can verify this by summing up all the three probabilities:

                            P ( X < 60 ) +  P ( 60 < X < 140 ) +  P ( X > 140 ) = 1

-Hence,  P ( X > 140 ) = 0.025

-Hence, from empirical rule we have