The Intelligence Quotient (IQ) test scores are normally distributed with a mean of 100 and a standard deviation of 15. i. What is the probability that a person would score 130 or more on the test?
ii. You enrolled in a class of 25 students. What is the probability that the class' average IQ exceeds 130?
A) 0.0228, 0.4987
B) 0.4772, 0.300
C) 0.4772, 0.0026
D) 0.0228, approximately zero

Question

Quizplus · Accepted Answer

i. Using the Z-score formula, we get:
z = (x - μ) / σ
z = (130 - 100) / 15
z = 2
Looking at the normal distribution table, the probability of getting a value of 2 or higher is 0.0228.
Therefore, the probability that a person would score 130 or more on the test is 0.0228.

ii. The distribution of sample means follows the Central Limit Theorem, where the mean of the sample means is equal to the population mean (100) and the standard deviation is equal to the population standard deviation divided by the square root of the sample size. 
Using the formula for the z-score for a sample mean:
z = (x̄ - μ) / (σ / √n)
z = (130 - 100) / (15 / √25)
z = 6.67
Looking at the normal distribution table, we can see that the probability of getting a value of 6.67 or higher is approximately zero. Therefore, the probability that the class' average IQ exceeds 130 is approximately zero.

The Intelligence Quotient (IQ) Test Scores Are Normally Distributed with a Mean