Respuesta :
Answer:
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, Â P ( X > 140 ) = 0.025
-Hence, from empirical rule we have
