Question

assume that adults have iq scores that are normally distributed with a mean of 103 and a standard deviation 21.1. 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 for the first quartile

The first quartile corresponds to the 25th percentile. Looking up the z - score in the standard normal distribution table (or using a calculator with a normal - distribution function), the z - score $z$ for the 25th percentile 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 from the original distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation. We know $\mu = 103$, $\sigma=21.1$, and $z=-0.674$. Rearranging the formula for $x$ gives $x=\mu + z\sigma$.

Step3: Calculate the first quartile

Substitute the values into the formula: $x = 103+(-0.674)\times21.1$. First, calculate $(-0.674)\times21.1=-14.2214$. Then, $x = 103-14.2214 = 88.7786$. Rounding to one decimal place, $x\approx88.8$.

Answer:

$88.8$