Question

Assume that adults have IQ scores that are normally distributed with a mean of 103.6 and a standard deviation of 16. Find the first quartile Q ₁, which is the IQ score separating the bottom 25% from the top 75%. (Hint: Draw a graph.)

The first quartile is

a) 87.1
b) 95.1
c) 103.6
d) 111.6

Answer 1

To find the first quartile, use the standard normal distribution to find the z-score corresponding to the 25th percentile, and then convert it back to an IQ score using the given mean and standard deviation.

Step-by-step explanation:

To find the first quartile, we need to find the IQ score that separates the bottom 25% from the top 75%. Since the IQ scores are normally distributed, we can use the standard normal distribution to find the z-score corresponding to the 25th percentile. The z-score can then be converted back to an IQ score using the given mean and standard deviation. The first quartile is the IQ score separating the bottom 25% from the top 75%.

Using the standard normal distribution table, the z-score corresponding to the 25th percentile is approximately -0.674. We can find the corresponding IQ score using the formula:

IQ score = (z-score * standard deviation) + mean

Plugging in the values, we get:

IQ score = (-0.674 * 16) + 103.6 = 87.1

Therefore, the first quartile Q₁, which is the IQ score separating the bottom 25% from the top 75%, is 87.1 (option a).