Question

people with z - scores of 2.5 or above on a certain aptitude test are sometimes classified as geniuses. if aptitude test scores have a mean of 100 and a standard deviation of 30 points, what is the minimum aptitude test score needed to be considered a genius? the minimum aptitude test score needed to be considered a genius is points. (type an integer or a decimal. do not round.)

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $z$ is the z - score, $x$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation. We want to find $x$, and we know that $z = 2.5$, $\mu=100$, and $\sigma = 30$.

Step2: Rearrange the formula to solve for $x$

Starting from $z=\frac{x - \mu}{\sigma}$, we can multiply both sides by $\sigma$: $z\sigma=x-\mu$. Then add $\mu$ to both sides to get $x=\mu + z\sigma$.

Step3: Substitute the given values

Substitute $\mu = 100$, $z = 2.5$, and $\sigma=30$ into the formula $x=\mu + z\sigma$. So $x=100+2.5\times30$.

Step4: Calculate the value of $x$

First, calculate $2.5\times30 = 75$. Then $x=100 + 75=175$.

Answer:

175