A population of values has a normal distribution with \mu=239.5 and \sigma=32.7. You intend to draw a random sample of size n=139. Find the probability that a sample of size n=139 is randomly selected with a mean greater than 235.9. P(M > 235.9) =? Write your answers as numbers accurate to 4 decimal places.

Question
Random variables
A population of values has a normal distribution with $$\displaystyle\mu={239.5}$$ and $$\displaystyle\sigma={32.7}$$. You intend to draw a random sample of size $$\displaystyle{n}={139}$$.
Find the probability that a sample of size n=139 is randomly selected with a mean greater than 235.9.
$$\displaystyle{P}{\left({M}{>}{235.9}\right)}=$$?

2020-10-26
From the provided information,
Mean $$\displaystyle{\left(\mu\right)}={239.5}$$
Standard deviation $$\displaystyle{\left(\sigma\right)}={32.7}$$
Sample size $$\displaystyle{\left({n}\right)}={139}$$
Let X be a random variable which represents the value score.
$$\displaystyle{X}\sim{N}{\left({239.5},{32.7}\right)}$$
The required probability that a sample of size $$\displaystyle{n}={139}$$ is randomly selected with a mean greater than 235.9 can be obtained as:
$$\displaystyle{P}{\left({M}{>}{235.9}\right)}={P}{\left({\frac{{{M}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}{>}{\frac{{{235.9}-{239.5}}}{{{\frac{{{32.7}}}{{\sqrt{{{139}}}}}}}}}\right)}$$
$$\displaystyle={P}{\left({Z}\succ{1.298}\right)}$$
$$\displaystyle={1}-{P}{\left({Z}{<}-{1.298}\right)}$$</span>
$$\displaystyle={1}-{0.0971}={0.9029}$$ (Using standard normal table)
Hence, the required probability is 0.9029.

Relevant Questions

A population of values has a normal distribution with $$\displaystyle\mu={239.5}$$ and $$\displaystyle\sigma={32.7}$$. You intend to draw a random sample of size $$\displaystyle{n}={139}$$.
Find the probability that a single randomly selected value is greater than 235.9.
$$\displaystyle{P}{\left({X}{>}{235.9}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={120.6}$$ and $$\displaystyle\sigma={48.5}$$. You intend to draw a random sample of size $$\displaystyle{n}={105}$$.
Find the probability that a sample of size $$\displaystyle{n}={105}$$ is randomly selected with a mean greater than 114.9.
$$\displaystyle{P}{\left({M}{>}{114.9}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={182.5}$$ and $$\displaystyle\sigma={49.4}$$. 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 greater than 169.7.
$$\displaystyle{P}{\left({M}{>}{169.7}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={73.1}$$ and $$\displaystyle\sigma={28.1}$$. You intend to draw a random sample of size $$\displaystyle{n}={131}$$.
Find the probability that a sample of size $$\displaystyle{n}={131}$$ is randomly selected with a mean greater than 69.7.
$$\displaystyle{P}{\left({M}{>}{69.7}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={49}$$ and $$\displaystyle\sigma={79.5}$$. You intend to draw a random sample of size $$\displaystyle{n}={84}$$.
Find the probability that a a sample of size $$\displaystyle{n}={84}$$ is randomly selected with a mean greater than 72.4.
$$\displaystyle{P}{\left({M}{>}{72.4}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={154.5}$$ and $$\displaystyle\sigma={96.1}$$. You intend to draw a random sample of size $$\displaystyle{n}={134}$$.
Find the probability that a sample of size $$\displaystyle{n}={134}$$ is randomly selected with a mean greater than 167.
$$\displaystyle{P}{\left({M}{>}{167}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={129.7}$$ and $$\displaystyle\sigma={7.7}$$. You intend to draw a random sample of size $$\displaystyle{n}={10}$$.
Find the probability that a sample of size $$\displaystyle{n}={10}$$ is randomly selected with a mean less than 130.9.
$$\displaystyle{P}{\left({M}{<}{130.9}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={13.7}$$ and $$\displaystyle\sigma={22}$$.
You intend to draw a random sample of size $$\displaystyle{n}={78}$$.
Find the probability that a sample of size $$\displaystyle{n}={78}$$ is randomly selected with a mean less than 11.5.
$$\displaystyle{P}{\left({M}{<}{11.5}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={116.3}$$ and $$\displaystyle\sigma={27.5}$$. You intend to draw a random sample of size $$\displaystyle{n}={249}$$.
Find the probability that a sample of size $$\displaystyle{n}={249}$$ is a randomly selected with a mean greater than 117.3.
$$\displaystyle{P}{\left(\overline{{{X}}}{>}{117.3}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={120.6}$$ and $$\displaystyle\sigma={48.5}$$. You intend to draw a random sample of size $$\displaystyle{n}={105}$$.
$$\displaystyle{P}{\left({X}{>}{114.9}\right)}=$$?