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.)

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Mathematics

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19 October, 05:19

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.)

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Answers (1)

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Carla Price · Answer 1

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) = 1

-Hence, P (X > 140) = 0.025

-Hence, from empirical rule we have