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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.)

User Poorya Pzm
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Answer:

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

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

c) P ( X > 140 ) = 0.025

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

User Vishal Shah
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