Question

question 6 of 10 a tennis ball machine fires balls a distance that is approximately normally distributed. the mean distance, μ, is unknown and the standard deviation is 1.2 feet. if 5% of balls go farther than 70 feet, find μ. μ = feet (round to 3 decimal places.)

Explanation:

Step1: Find the z - score

If 5% of balls go farther than 70 feet, then the area to the right of 70 under the normal - distribution curve is 0.05. The area to the left of 70 is $1 - 0.05=0.95$. Looking up 0.95 in the standard normal distribution table (z - table), the corresponding z - score $z$ is approximately 1.645.

Step2: Use the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x = 70$ (the value from the problem), $\mu$ is the mean (which we want to find), and $\sigma = 1.2$ (the standard deviation).
Substitute the known values into the formula: $1.645=\frac{70 - \mu}{1.2}$.

Step3: Solve for $\mu$

First, multiply both sides of the equation by 1.2: $1.645\times1.2=70 - \mu$.
$1.974 = 70-\mu$.
Then, add $\mu$ to both sides: $\mu+1.974 = 70$.
Finally, subtract 1.974 from both sides: $\mu=70 - 1.974$.
$\mu = 68.026$.

Answer:

$68.026$