I need help with this question please?

Question

Assume that adults have IQ scores that are normally distributed with a mean of 101.5 and a standard deviation 21.4 . Find the first quartile Q_(1), which is the IQ score separating the bottom 25% from the top 75%. (Hint: Draw a graph.)

Answer 1

Hi Dylan,

I'm actually not sure if you need to draw a graph. This problem involves the classic equation in introductory statistics:

z=(x-mu)/sigma

where x=value you are looking for

mu=population mean

sigma=standard deviation

z=z-score explained below

Now, recall that the first quartile is equivalent to the 25th percentile. Navigate to a z-table online and search the interior for the decimal equivalent of the 25th percentile, 0.25. The two closest values on my z-table are 0.2514 and 0.2483. Looking at the column on the left and row on top, this corresponds to:

z= -0.675

Now, we can substitute into our classic equation

z=(x-mu)/sigma

z= -0.675

x=x, the IQ we are looking for

mu=101.5

sigma=21.4

-0.675= (x-101.5)/21.4

Multiply both sides by 21.4

-14.445=x-101.5

Add 101.5:

x=87.055, rounded down,

x=87

I hope this helps.