# A population of values has a normal distribution with \mu = 116.5 and \sigma = 63.7. You intend to draw a random sample of size n = 244. Find the probability that a sample of size n = 244 is randomly selected with a mean between 104.7 and 112.8. P(104.7 < M < 112.8)=? Write your answers as numbers accurate to 4 decimal places.

Question
Random variables
A population of values has a normal distribution with $$\displaystyle\mu={116.5}$$ and $$\displaystyle\sigma={63.7}$$. You intend to draw a random sample of size $$\displaystyle{n}={244}$$.
Find the probability that a sample of size $$\displaystyle{n}={244}$$ is randomly selected with a mean between 104.7 and 112.8.
$$\displaystyle{P}{\left({104.7}{<}{M}{<}{112.8}\right)}=$$</span>?

2020-11-30
A population of values has a normal distribution with $$\displaystyle\mu={116.5}$$ and $$\displaystyle\sigma={63.7}$$.
A random sample of size $$\displaystyle{\left({n}\right)}={244}$$ is drawn.
We need to find the probability that a sample of size $$\displaystyle{n}={244}$$ is randomly selected with a mean(M) between 104.7 and 112.8.
$$\displaystyle{M}\sim{N}{\left({116.5},{\frac{{{63.7}^{{{2}}}}}{{{n}}}}\right)}$$ and $$\displaystyle{n}={244}$$
So, $$\displaystyle{Z}={\frac{{\sqrt{{{n}}}{\left({M}-{116.5}\right)}}}{{{63.7}}}}\sim{N}{\left({0},{1}\right)}$$
Thus, $$\displaystyle{P}{\left[{104.7}{<}{M}{<}{112.8}\right]}$$</span>
$$\displaystyle={P}{\left[{\frac{{\sqrt{{{n}}}{\left({104.7}-{116.5}\right)}}}{{{63.7}}}}{<}{\frac{{\sqrt{{{n}}}{\left({M}-{116.5}\right)}}}{{{63.7}}}}{<}{\frac{{\sqrt{{{n}}}{\left({112.8}-{116.5}\right)}}}{{{63.7}}}}\right]}$$</span>
$$\displaystyle={P}{\left[-{2.893}{<}{Z}{<}-{0.9073}\right]}$$</span>
$$\displaystyle={P}{\left[{Z}\leq-{0.9073}\right]}-{P}{\left[{Z}\leq-{2.8934}\right]}$$
$$\displaystyle={0.1821}-{0.0019}={0.1802}$$
So, the probability that a sample of size $$\displaystyle{n}={244}$$ is randomly selected with a mean between 104.7 and 112.8 is 0.1802.

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