assume the random variable x is normally distributed with mean $mu = 50$ and standard deviation $sigma = 7$…

Explanation:

Step1: Find the z - score

We use the standard normal distribution table (z - table). The 78th percentile means that the area to the left of the z - score is 0.78. Looking up in the z - table, the z - score corresponding to an area of 0.78 is approximately $z = 0.77$.

Step2: Use the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the original normal distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation. We want to solve for $x$. Rearranging the formula gives $x=\mu + z\sigma$. We know that $\mu = 50$, $\sigma=7$, and $z = 0.77$. Substituting these values into the formula: $x=50+0.77\times7$ $x = 50 + 5.39$ $x=55.39$

Answer:

$55.39$