Question

Assume that adults have IQ score that are normally distributed with a mean of 99.4 and a standard deviation 16.7 find the first Quartile Q1 which is the IQ score separate in the bottom 25% from the top 75%.

Answer 1

First quartile (Q1) IQ score that separates the bottom 25% from the top 75% is approximately 88.5

Explanation:

To find the first quartile (Q1) of the IQ scores, which separates the bottom 25% from the top 75%, you can use the standard normal distribution (Z-score) and then convert it back to the original IQ scale.

First, you need to find the Z-score for the 25th percentile, which corresponds to Q1. You can use the standard normal distribution table or a calculator for this purpose. The Z-score for the 25th percentile is approximately -0.6745.

Now, you can use the Z-score formula to find the actual IQ score (X) associated with this Z-score:

Z = (X - μ) / σ

Plugging in the values:

-0.6745 = (X - 99.4) / 16.7

Now, solve for X:

X = (-0.6745 * 16.7) + 99.4

X ≈ 88.5

So, the first quartile (Q1) IQ score that separates the bottom 25% from the top 75% is approximately 88.5. This means that 25% of the adult population has an IQ score below 88.5, while 75% have an IQ score above 88.5.