# A population of values has a normal distribution with \mu=192.6 and \sigma=34.4.

Question
Random variables

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 sample of size $$\displaystyle{n}={173}$$ is randomly selected with a mean less than 186.1.
$$\displaystyle{P}{\left({M}{<}{186.1}\right)}=$$?

2021-02-26

Step 1
Given:
$$\displaystyle\mu={192.6}$$,
$$\displaystyle\sigma={34.4}$$,
$$\displaystyle{n}={173}$$.
The Z-score follows standard normal distribution.
Step 2
$$\displaystyle{P}{\left[{X}{<}{186.1}\right]}={P}{\left[{\frac{{{X}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}{<}{\frac{{{186.1}-{192.6}}}{{{\frac{{{34.4}}}{{\sqrt{{{173}}}}}}}}}\right]}$$
$$\displaystyle={P}{\left[{Z}{<}-{2.49}\right]}={0.0066}$$ (Use standard normal table)
The probability that a sample of size $$\displaystyle{n}={173}$$ is randomly selected with a mean less than 186.1 is 0.0066.

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