Question

A population of values has a normal distribution with \(\displaystyle\mu={129.7}\) and \(\displaystyle\sigma={7.7}\). You intend to draw a random sample of size \(\displaystyle{n}={10}\).
Find the probability that a sample of size \(\displaystyle{n}={10}\) is randomly selected with a mean less than 130.9.
\(\displaystyle{P}{\left({M}{<}{130.9}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.

Answer 1

Step 1
Given Information:
\(\displaystyle\mu={129.7}\)
\(\displaystyle\sigma={7.7}\)
\(\displaystyle{n}={10}\)
Step 2
The probability that a sample of size \(\displaystyle{n}={10}\) is randomly selected with a mean less than 130.9.
\(\displaystyle{z}={\frac{{{x}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}={\frac{{{130.9}-{129.7}}}{{{\frac{{{7.7}}}{{\sqrt{{{n}}}}}}}}}={0.4928}\)
By referring to the z distribution, the p-value at \(\displaystyle{z}={0.492}\) is 0.311
Therefore, The probability that a sample of size \(\displaystyle{n}={10}\) is randomly selected with a mean less than 130.9 is 0.311