Question

assume that adults have iq scores that are normally distributed with a mean of 104.8 and a standard deviation 19.5. 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
(type an integer or decimal rounded to one decimal place as needed.)

Explanation:

Step1: Find the z - score corresponding to the first quartile

The first quartile \(Q_1\) has an area of \(0.25\) to its left. Using a standard normal table (or a calculator with a normal - distribution function), the \(z\) - score \(z\) such that \(P(Z\lt z)=0.25\) is approximately \(z=- 0.674\).

Step2: Use the z - score formula to find \(Q_1\)

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 104.8\) (mean), \(\sigma = 19.5\) (standard deviation), and \(z=-0.674\).
We want to solve for \(x\) (which is \(Q_1\)). Rearranging the formula gives \(x=\mu+z\sigma\).
Substitute the values: \(x = 104.8+(-0.674)\times19.5\).
First, calculate \((-0.674)\times19.5=-13.143\).
Then, \(x = 104.8-13.143\).

Answer:

\(91.7\)