# A population of values has a normal distribution with \mu=77 and \sigma=32.2. You intend to draw a random sample of size n=15 Find the probability that a sample of size n=15 is randomly selected with a mean between 59.5 and 98.6. P(59.5 < \bar{X} < 98.6) =? Write your answers as numbers accurate to 4 decimal places.

Question
Random variables
A population of values has a normal distribution with $$\displaystyle\mu={77}$$ and $$\displaystyle\sigma={32.2}$$. You intend to draw a random sample of size $$\displaystyle{n}={15}$$
Find the probability that a sample of size $$\displaystyle{n}={15}$$ is randomly selected with a mean between 59.5 and 98.6. $$\displaystyle{P}{\left({59.5}{<}\overline{{{X}}}{<}{98.6}\right)}=$$</span>?

2020-12-23
From the given information, $$\displaystyle\mu={77}$$, $$\displaystyle\sigma={32.2}$$ and the sample size=15.
Consider,
$$\displaystyle{P}{\left({59.5}{<}\overline{{{X}}}{<}{98.6}\right)}={P}{\left({\frac{{{59.5}-\mu}}{{{\left({\frac{{\sigma}}{{\sqrt{{{n}}}}}}\right)}}}}{<}{\frac{{\overline{{{X}}}-\mu}}{{{\left({\frac{{\sigma}}{{\sqrt{{{n}}}}}}\right)}}}}{<}{\frac{{{98.6}-\mu}}{{{\left({\frac{{\sigma}}{{\sqrt{{{n}}}}}}\right)}}}}\right)}$$</span>
$$\displaystyle={P}{\left({\frac{{{59.5}-\mu}}{{{\left({\frac{{{32.2}}}{{\sqrt{{{15}}}}}}\right)}}}}{<}{z}{<}{\frac{{{98.6}-{77}}}{{{\left({\frac{{{32.2}}}{{\sqrt{{{15}}}}}}\right)}}}}\right)}$$</span>
$$\displaystyle={P}{\left(-{2.10488}{<}{z}{<}{2.59803}\right)}$$</span>
$$\displaystyle={P}{\left({z}{<}{2.59803}\right)}-{P}{\left({z}{<}-{2.10488}\right)}$$</span>
$$\displaystyle={0.995312}-{0.017651}={0.9777}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{F}{r}{o}{m}\ {t}{h}{e}\ {E}{x}{c}{e}{l}\ {f}{u}{n}{c}{t}{i}{o}{n}\backslash={N}{O}{R}{M}.{D}{I}{S}{T}{\left(-{2.10488},{0},{1},{T}{R}{U}{E}\right)}\backslash={N}{O}{R}{M}.{D}{I}{S}{T}{\left({2.59803},{0},{1},{T}{R}{U}{E}\right)}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$
Thus, the probability that a sample of size $$\displaystyle{n}={15}$$ is randomly selected with a mean between 59.5 and 98.6 is 0.9777.

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