Question

complete the statement. round your answers to the nearest thousandth.
in a population that is normally distributed with mean 40 and standard deviation 11,
the outside 50% of the values are those less than or greater than.

Explanation:

Step1: Find the z - scores

Since we want the outside 50% of the values, the middle 50% of the values are between two z - scores. The area in the middle is \(0.5\), so the area in each tail is \(\frac{1 - 0.5}{2}=0.25\).
Looking up the z - score for an area of \(0.25\) in the standard normal table (left - tail area), \(z_{1}=- 0.674\) and \(z_{2}=0.674\) (because of the symmetry of the normal distribution).

Step2: Use the z - score formula \(x=\mu+z\sigma\)

Given \(\mu = 40\) and \(\sigma=11\)
For \(z=-0.674\):
\(x_{1}=\mu + z_{1}\sigma=40+(-0.674)\times11\)
\(x_{1}=40 - 7.414=32.586\)
For \(z = 0.674\):
\(x_{2}=\mu+z_{2}\sigma=40 + 0.674\times11\)
\(x_{2}=40+7.414 = 47.414\)

Answer:

The outside 50% of the values are those less than \(32.586\) or greater than \(47.414\)