# A population of values has a normal distribution with

Question
Random variables

A population of values has a normal distribution with $$\displaystyle\mu={200.7}$$ and $$\displaystyle\sigma={10}$$. You intend to draw a random sample of size $$\displaystyle{n}={178}$$
Find the probability that a sample of size $$\displaystyle{n}={178}$$ is randomly selected with a mean less than 198.6.
$$\displaystyle{P}{\left({M}{<}{198.6}\right)}=$$?

2021-01-20

Step 1
From the provided information,
Mean $$\displaystyle{\left(\mu\right)}={200.7}$$
Standard deviation $$\displaystyle{\left(\sigma\right)}={10}$$
Let X be a random variable which represents the value.
$$\displaystyle{X}\sim{N}{\left({200.7},{10}\right)}$$
Sample size $$\displaystyle{\left({n}\right)}={178}$$
Step 2
The required probability that a sample of size $$\displaystyle{n}={178}$$ is randomly selected with a mean less than 198.6 can be obtained as:
$$\displaystyle{P}{\left({M}{<}{198.6}\right)}={P}{\left({\frac{{{M}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}{<}{\frac{{{198.6}-{200.7}}}{{{\frac{{{10}}}{{\sqrt{{{178}}}}}}}}}\right)}$$
$$\displaystyle={P}{\left({Z}{<}-{2.802}\right)}={0.0025}$$ (Using standard normal table)
Thus, the required probability is 0.0025.

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