Question

A population of values has a normal distribution with \(\displaystyle\mu={192.6}\) and \(\displaystyle\sigma={34.4}\). You intend to draw a random sample of size \(\displaystyle{n}={173}\).
Find the probability that a single randomly selected value is less than 186.1.
\(\displaystyle{P}{\left({X}{<}{186.1}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.

Answer 1

Step 1
Given:
\(\displaystyle\mu={192.6}\),
\(\displaystyle\sigma={34.4}\),
\(\displaystyle{n}={173}\).
The Z-score follows standard normal distribution.
Step 2
\(\displaystyle{P}{\left[{X}{<}{186.1}\right]}={P}{\left[{\frac{{{X}-\mu}}{{\sigma}}}{<}{\frac{{{186.1}-{192.6}}}{{{34.4}}}}\right]}\)
\(\displaystyle={P}{\left[{Z}{<}-{0.19}\right]}={0.4247}\) (Use standard normal table)
The probability that a single randomly selected value is less than 186.1 is 0.4247.