Question

In a normal distribution of IQs with a mean of 100 and a standard deviation of 15, the middle 50% of IQs fall between approximately:

a) 85 and 115
b) 90 and 110
c) 75 and 125
d) 80 and 120

Answer 1

In a normal distribution with a mean of 100 and a standard deviation of 15, the middle 50% of IQs fall between approximately 102 and 112.

Step-by-step explanation:

In a normal distribution with a mean of 100 and a standard deviation of 15, the middle 50% of IQs fall between the 25th and 75th percentiles.

To find these values, we use the z-score formula:

z = (x - mean) / standard deviation

For the lower bound:

z = (85 - 100) / 15 = -1

For the upper bound:

z = (115 - 100) / 15 = 1

Using a z-table, we find the corresponding percentiles for z = -1 and z = 1, which are approximately 0.1587 and 0.8413, respectively.

The middle 50% of IQs fall between the 16th percentile (0.1587) and the 84th percentile (0.8413).

Converting these percentiles back to IQ values:

Lower bound: IQ = (0.1587 * 15) + 100 = 102.38 ≈ 102

Upper bound: IQ = (0.8413 * 15) + 100 = 112.12 ≈ 112

Therefore, the middle 50% of IQs fall between approximately 102 and 112.