Ask question

# A population of values has a normal distribution with \mu=197 and \sigma=68. You intend to draw a random sample of size n=181 Find the probability that a sample of size n=181 is randomly selected with a mean between 198.5 and 205.6. P(198.5 < M < 205.6) =? Write your answers as numbers accurate to 4 decimal places.

Question
Random variables
asked 2021-01-28
A population of values has a normal distribution with $$\displaystyle\mu={197}$$ and $$\displaystyle\sigma={68}$$. You intend to draw a random sample of size $$\displaystyle{n}={181}$$
Find the probability that a sample of size $$\displaystyle{n}={181}$$ is randomly selected with a mean between 198.5 and 205.6.
$$\displaystyle{P}{\left({198.5}{<}{M}{<}{205.6}\right)}=$$</span>?
Write your answers as numbers accurate to 4 decimal places.

## Answers (1)

2021-01-29
Step 1
From the provided information,
Mean $$\displaystyle{\left(\mu\right)}={197}$$
Standard deviation $$\displaystyle{\left(\sigma\right)}={68}$$
Let X be a random variable which represents the value.
$$\displaystyle{X}\sim{N}{\left({197},{68}\right)}$$
Step 2
Sample size $$\displaystyle{\left({n}\right)}={181}$$
The required probability that a sample of size $$\displaystyle{n}={181}$$ is randomly selected with a mean between 198.5 and 205.6 $$\displaystyle{P}{\left({198.5}{<}{M}{<}{205.6}\right)}$$</span> can be obtained as:
$$\displaystyle{P}{\left({198.5}{<}{M}{<}{205.6}\right)}={P}{\left({\frac{{{198.5}-{197}}}{{{\frac{{{68}}}{{\sqrt{{{181}}}}}}}}}{<}{\frac{{{M}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{181}}}}}}}}}{<}{\frac{{{205.6}-{197}}}{{{\frac{{{68}}}{{\sqrt{{{181}}}}}}}}}\right)}$$</span>
$$\displaystyle={P}{\left({0.297}{<}{Z}{<}{1.701}\right)}$$</span>
$$\displaystyle={P}{\left({Z}{<}{1.701}\right)}-{P}{\left({Z}{<}{0.297}\right)}$$</span>
$$\displaystyle={0.9555}-{0.6168}={0.3387}$$ (Using standard normal table)
Thus, the required probability is 0.3387.

### Relevant Questions

asked 2020-11-22
A population of values has a normal distribution with $$\displaystyle\mu={197}$$ and $$\displaystyle\sigma={68}$$. You intend to draw a random sample of size $$\displaystyle{n}={181}$$
Find the probability that a single randomly selected value is between 198.5 and 205.6.
$$\displaystyle{P}{\left({198.5}{<}{X}{<}{205.6}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-01-04
A population of values has a normal distribution with $$\displaystyle\mu={198.8}$$ and $$\displaystyle\sigma={69.2}$$. You intend to draw a random sample of size $$\displaystyle{n}={147}$$.
Find the probability that a sample of size $$\displaystyle{n}={147}$$ is randomly selected with a mean between 184 and 205.1.
$$\displaystyle{P}{\left({184}{<}{M}{<}{205.1}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.
asked 2020-10-26
A population of values has a normal distribution with $$\displaystyle\mu={198.8}$$ and $$\displaystyle\sigma={69.2}$$. You intend to draw a random sample of size $$\displaystyle{n}={147}$$.
Find the probability that a single randomly selected value is between 184 and 205.1.
$$\displaystyle{P}{\left({184}{<}{X}{<}{205.1}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-02-05
A population of values has a normal distribution with $$\displaystyle\mu={181}$$ and $$\displaystyle\sigma={41.8}$$. You intend to draw a random sample of size $$\displaystyle{n}={144}$$.
Find the probability that a single randomly selected value is between 176.5 and 183.1.
$$\displaystyle{P}{\left({176.5}{<}{M}{<}{183.1}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-02-05
A population of values has a normal distribution with $$\displaystyle\mu={181}$$ and $$\displaystyle\sigma={41.8}$$. You intend to draw a random sample of size $$\displaystyle{n}={144}$$.
Find the probability that a single randomly selected value is between 176.5 and 183.1.
$$\displaystyle{P}{\left({176.5}{<}{X}{<}{183.1}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.
asked 2020-12-06
A population of values has a normal distribution with $$\displaystyle\mu={13.2}$$ and $$\displaystyle\sigma={5}$$. You intend to draw a random sample of size $$\displaystyle{n}={60}$$.
Find the probability that a sample of size $$\displaystyle{n}={60}$$ is randomly selected with a mean between 11.5 and 14.
$$\displaystyle{P}{\left({11.5}{<}{M}{<}{14}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.
asked 2020-11-29
A population of values has a normal distribution with $$\displaystyle\mu={116.5}$$ and $$\displaystyle\sigma={63.7}$$. You intend to draw a random sample of size $$\displaystyle{n}={244}$$.
Find the probability that a sample of size $$\displaystyle{n}={244}$$ is randomly selected with a mean between 104.7 and 112.8.
$$\displaystyle{P}{\left({104.7}{<}{M}{<}{112.8}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-01-05
A population of values has a normal distribution with $$\displaystyle\mu={99.6}$$ and $$\displaystyle\sigma={35.1}$$. You intend to draw a random sample of size $$\displaystyle{n}={84}$$.
Find the probability that a sample of size $$\displaystyle{n}={84}$$ is randomly selected with a mean between 98.5 and 100.7.
$$\displaystyle{P}{\left({98.5}{<}\overline{{{X}}}{<}{100.7}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.
asked 2020-12-22
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. $$\displaystyle{P}{\left({59.5}{<}\overline{{{X}}}{<}{98.6}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-02-05
A population of values has a normal distribution with $$\displaystyle\mu={133.5}$$ and $$\displaystyle\sigma={5.2}$$. You intend to draw a random sample of size $$\displaystyle{n}={230}$$.
Find the probability that a sample of size $$\displaystyle{n}={230}$$ is randomly selected with a mean between 133.6 and 134.1.
$$\displaystyle{P}{\left({133.6}{<}\overline{{{x}}}{<}{134.1}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.
...