Question

a set of data items is normally distributed with a mean of 700 and a standard deviation of 10. find the data item in this distribution that corresponds to the given z - score. z = 1.5 the data item that corresponds to z = 1.5 is 760. (type an integer or a decimal.)

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 item, $\mu$ is the mean, and $\sigma$ is the standard deviation. We need to solve for $x$.

Step2: Rearrange the formula 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 given values

We are given that $\mu = 700$, $z = 1.5$, and $\sigma=10$. Substitute these values into the formula $x=\mu + z\sigma$: $x=700+1.5\times10$.

Step4: Calculate the value of $x$

First, calculate $1.5\times10 = 15$. Then, $x=700 + 15=715$.

Answer:

715