# A population of values has a normal distribution with \mu=26.8 and \sigma=33.8. You intend to draw a random sample of size n=89. Find the probability that a sample of size n=89 is randomly selected with a mean between 17.1 and 25. P(17.1 < M < 25) =? Write your answers as numbers accurate to 4 decimal places.

Question
Random variables
A population of values has a normal distribution with $$\displaystyle\mu={26.8}$$ and $$\displaystyle\sigma={33.8}$$. You intend to draw a random sample of size $$\displaystyle{n}={89}$$.
Find the probability that a sample of size $$\displaystyle{n}={89}$$ is randomly selected with a mean between 17.1 and 25.
$$\displaystyle{P}{\left({17.1}{<}{M}{<}{25}\right)}=$$</span>?

2020-12-16
Step 1
It is given that a population of values distributed normally with mean 26.8 and standard deviation 33.8. The sample size is 89.
Step 2
Calculate the probability that a sample of size $$\displaystyle{n}={89}$$ is randomly selected with a mean between 17.1 and 25 is as follows:
$$\displaystyle{P}{\left({17.1}{<}{X}{<}{25}\right)}={P}{\left({\frac{{{17.1}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}{<}{\frac{{{X}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}{<}{\frac{{{25}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}\right)}$$</span>
$$\displaystyle={P}{\left({\frac{{{17.1}-{26.8}}}{{{\frac{{{33.8}}}{{\sqrt{{{89}}}}}}}}}{<}{Z}{<}{\frac{{{25}-{26.8}}}{{{\frac{{{33.8}}}{{\sqrt{{{89}}}}}}}}}\right)}$$</span>
$$\displaystyle{P}{\left(-{\frac{{{9.7}}}{{{3.58279}}}}{<}{Z}{<}{\frac{{{1.8}}}{{{3.58279}}}}\right)}$$</span>
$$\displaystyle={P}{\left(-{2.707}{<}{Z}{<}-{0.502}\right)}$$</span>
$$\displaystyle={P}{\left({Z}{<}-{0.502}\right)}-{P}{\left({Z}{<}-{2.707}\right)}{\left({b}{e}{g}\in{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}{U}{\sin{{g}}}\ {E}{x}{c}{e}{l}\ {f}{u}{n}{c}{t}{i}{o}{n}{s},\backslash{P}{\left({Z}{<}-{2.707}\right)}={0.0034}\backslash{P}{\left({Z}{<}-{0.502}\right)}={0.3078}{e}{n}{d}{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}$$</span>
$$\displaystyle={0.3078}-{0.0034}={0.3044}$$
Therefore, the value of $$\displaystyle{P}{\left({17.1}{<}{M}{<}{25}\right)}={0.3044}$$</span>.

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