Question

assume that adults have iq scores that are normally distributed with a mean of 100.4 and a standard deviation 20.6. find the first - quartile q1, 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

We know that we want to find the value that separates the bottom 25% (or 0.25) from the top 75%. Looking up 0.25 in the standard normal distribution table (z - table), the z - score corresponding to a cumulative probability of 0.25 is approximately $z=- 0.674$.

Step2: Use the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value we want to find, $\mu$ is the mean, and $\sigma$ is the standard deviation. We are given $\mu = 100.4$ and $\sigma=20.6$, and we found $z=-0.674$. Rearranging the formula for $x$ gives $x=\mu + z\sigma$.
Substitute the values: $x = 100.4+( - 0.674)\times20.6$.
First, calculate $( - 0.674)\times20.6=-0.674\times20.6=-13.8844$.
Then, $x = 100.4-13.8844 = 86.5156$.
Rounding to one decimal place, $x\approx86.5$.

Answer:

$86.5$