The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find the mean of this sample is between 95 and 105?
A) 0.0182
B) 0.9818
C) 1.00
D) Very unlikely

Question

Quizplus · Accepted Answer

Since the sample size is greater than 30, we can use the Central Limit Theorem, which states that the sample means are normally distributed with a mean of the population and a standard deviation of (population standard deviation / square root of sample size).
Thus, the mean of the distribution of sample means is 100 and the standard deviation is 15/sqrt(50) = 2.12.
We can standardize this distribution by using the z-score formula: z = (x - mean) / standard deviation.
Therefore, 
z for x = 95: z = (95-100)/2.12 = -2.36
z for x = 105: z = (105-100)/2.12 = 2.36
Using the z-table, we can find the area under the normal distribution curve between these two z-scores:
Area = 0.4908 - 0.0090 = 0.4818
Therefore, the probability that we could select a sample of 50 adults and find the mean of this sample is between 95 and 105 is 0.4818 or 48.18%. Option B is correct.

The Intelligence Quotient (IQ) Test Scores for Adults Are Normally