A population of values has a normal distribution with \mu = 192.3 and \sigma = 66.5. 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 less than 185.4. P(M<185.4) =? Write your answers as numbers accurate to 4 decimal places.

A population of values has a normal distribution with \mu = 192.3 and \sigma = 66.5. 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 less than 185.4. P(M<185.4) =? Write your answers as numbers accurate to 4 decimal places.

Question
Random variables
asked 2020-10-27
A population of values has a normal distribution with \(\displaystyle\mu={192.3}\) and \(\displaystyle\sigma={66.5}\). 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 less than 185.4.
\(\displaystyle{P}{\left({M}{<}{185.4}\right)}=\)</span>?
Write your answers as numbers accurate to 4 decimal places.

Answers (1)

2020-10-28
The known values are,
\(\displaystyle\mu={192.3}\),
\(\displaystyle\sigma={66.5}\),
\(\displaystyle{n}={15}\)
The probability that a sample size \(\displaystyle{n}={15}\) is randomly selected with a mean less than 185.4 is,
\(\displaystyle{P}{\left({M}{<}{185.4}\right)}={P}{\left({\frac{{{M}-\mu}}{{\frac{\sigma}{\sqrt{{{n}}}}}}}{<}{\frac{{{185.4}-{192.3}}}{{\frac{{66.5}}{\sqrt{{{15}}}}}}}\right)}\)</span>
\(\displaystyle={P}{\left({z}{<}-{0.402}\right)}\)</span>
\(\displaystyle={\left(={N}{O}{R}{M}{D}{I}{S}{T}{\left(-{0.402}\right)}\right)}{\left({b}{e}{g}\in{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}{U}{\sin{{g}}}\ {t}{h}{e}\ {E}{x}{c}{e}{l}\ {f}{\quad\text{or}\quad}\mu{l}{a}\backslash{\left(={N}{O}{R}{M}{D}{I}{S}{T}{\left({z}\right)}\right)}{e}{n}{d}{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}\)
\(\displaystyle={0.343842}\approx{0.3438}\)
Therefore, the required probability is, 0.3438
0

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