# A population of values has a normal distribution with \mu=13.2 and \sigma=5. You intend to draw a random sample of size n=60. Find the probability that a sample of size n=60 is randomly selected with a mean between 11.5 and 14. P(11.5 < M < 14) =? Write your answers as numbers accurate to 4 decimal places.

Question
Random variables
A population of values has a normal distribution with $$\displaystyle\mu={13.2}$$ and $$\displaystyle\sigma={5}$$. You intend to draw a random sample of size $$\displaystyle{n}={60}$$.
Find the probability that a sample of size $$\displaystyle{n}={60}$$ is randomly selected with a mean between 11.5 and 14.
$$\displaystyle{P}{\left({11.5}{<}{M}{<}{14}\right)}=$$</span>?

2020-12-07
The probability that mean value is between 11.5 and 14 is 0.8884, which is obtained below:
$$\displaystyle{P}{\left({x}{1}{<}\overline{{{X}}}{<}{x}{2}\right)}={P}{\left({\frac{{{x}{1}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}{<}{z}{<}{\frac{{{x}{2}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}\right)}$$</span>
$$\displaystyle{P}{\left({11.5}{<}\overline{{{X}}}{<}{14}\right)}={P}{\left({\frac{{{11.5}-{13.2}}}{{{\frac{{{5}}}{{\sqrt{{{60}}}}}}}}}{<}{z}{<}{\frac{{{14}-{13.2}}}{{{\frac{{{5}}}{{\sqrt{{{60}}}}}}}}}\right)}$$</span>
$$\displaystyle={P}{\left(-{2.636}{<}{z}{<}{1.240}\right)}$$</span>
$$\displaystyle={P}{\left({z}{<}{1.240}\right)}-{P}{\left({z}{<}-{2.636}\right)}{\left[{U}{s}{e}\ {s}{\tan{{d}}}{a}{r}{d}\ {\left\|{a}\right\|}{l}\ {t}{a}{b}\le\right]}$$</span>
=0.89251-0.00415=0.8884.

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