# A population of values has a normal distribution with mean 18.6 and standard deviation 57. If a random sample of size n=25 is selected, Find the probability that a sample of size n=25 is randomly selected with a mean greater than 17.5 round your answer to four decimal places. P=?

Question
Random variables
A population of values has a normal distribution with mean 18.6 and standard deviation 57. If a random sample of size $$\displaystyle{n}={25}$$ is selected,
Find the probability that a sample of size $$\displaystyle{n}={25}$$ is randomly selected with a mean greater than 17.5 round your answer to four decimal places.
P=?

2020-11-09
A population of values has a normal distribution with mean 18.6 and standard deviation 57, sample size is $$\displaystyle{n}={25}$$. The value of x is 17.5. The z-score is,
$$\displaystyle{z}={\frac{{{x}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}$$
$$\displaystyle={\frac{{{17.5}-{18.6}}}{{{\frac{{{57}}}{{\sqrt{{{25}}}}}}}}}$$
$$\displaystyle={\frac{{-{1.1}}}{{{11.4}}}}=-{0.0965}$$
The area to the right of $$\displaystyle{z}=−{0.0965}$$ under the standard normal curve is $$\displaystyle{P}{\left({z}{>}−{0.0965}\right)}={1}−{P}{\left({z}{<}−{0.0965}\right)}$$</span>.
The probability of z less than –0.0965 can be obtained using the excel formula “=NORM.S.DIST(–0.0965,TRUE)”. The probability value is 0.4616.
The probability that a sample of size $$\displaystyle{n}={25}$$ is randomly selected with a mean greater than 17.5 is,
$$\displaystyle{P}{\left({z}\succ{0.0965}\right)}={1}-{P}{\left({z}{<}-{0.0965}\right)}$$</span>
$$\displaystyle={1}-{0.4616}={0.5384}$$
Thus, the probability that a sample of size $$\displaystyle{n}={25}$$ is randomly selected with a mean greater than 17.5 is 0.5384.

### Relevant Questions

A population of values has a normal distribution with mean =136.4 and standard deviation =30.2. A random sample of size $$\displaystyle{n}={158}$$ is drawn.
Find the probability that a sample of size $$\displaystyle{n}={158}$$ is randomly selected with a mean greater than 135. Round your answer to four decimal places.
$$\displaystyle{P}{\left({M}{>}{135}\right)}=$$?
A population of values has a normal distribution with mean =136.4 and standard deviation =30.2. A random sample of size $$\displaystyle{n}={158}$$ is drawn.
Find the probability that a single randomly selected value is greater than 135. Roung your answer to four decimal places.
$$\displaystyle{P}{\left({X}{>}{135}\right)}=$$?
A population of values has a normal distribution with mean = 37.4 and standard deviation 77.4. If a random sample of size $$\displaystyle{n}={15}$$ is selected,
Find the probability that a single randomly selected value is greater than 53.4. Round your answer to four decimals.
$$\displaystyle{P}{\left({X}{>}{53.4}\right)}=$$?
A population of values has a normal distribution with mean 191.4 and standard deviation of 69.7. A random sample of size $$\displaystyle{n}={153}$$ is drawn.
Find the probability that a sample of size $$\displaystyle{n}={153}$$ is randomly selected with a mean between 188 and 206.6. Round your answer to four decimal places.
P=?
A population of values has a normal distribution with mean 191.4 and standard deviation of 69.7. A random sample of size $$\displaystyle{n}={153}$$ is drawn.
Find the probability that a single randomly selected value is between 188 and 206.6 round your answer to four decimal places.
P=?
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 $$\displaystyle\mu={49}$$ 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 $$\displaystyle\mu={182.5}$$ 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 $$\displaystyle\mu={239.5}$$ and $$\displaystyle\sigma={32.7}$$. You intend to draw a random sample of size $$\displaystyle{n}={139}$$.
$$\displaystyle{P}{\left({M}{>}{235.9}\right)}=$$?
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.
$$\displaystyle{P}{\left({M}{>}{167}\right)}=$$?