Question

# A population of values has a normal distribution with \mu=129.7 and \sigma=7.7. You inte

Random variables

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.
$$\displaystyle{P}{\left({M}{<}{130.9}\right)}=$$?

2020-11-09
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