# A population of values has a normal distribution with \mu = 133.5 and \sigma = 5.2. You intend to draw a random sample of size n = 230. Find the probability that a sample of size n = 230 is randomly selected with a mean between 133.6 and 134.1. P(133.6<\bar{x}<134.1)=? Write your answers as numbers accurate to 4 decimal places.

Question
Random variables
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.
$$\displaystyle{P}{\left({133.6}{<}\overline{{{x}}}{<}{134.1}\right)}=$$</span>?

2021-02-06
Given Information:
$$\displaystyle\mu={133.5}$$ and $$\displaystyle\sigma={5.2}$$
$$\displaystyle{n}={230}$$
To find the probability that a sample of size $$\displaystyle{n}={230}$$ is randomly selected with a mean between 133.6 and 134.1:
Based on the concept of Central limit theorem, if a random sample of size 'n' is drawn from a population with mean $$\displaystyle\mu$$ and standard deviation $$\displaystyle\sigma$$, the sampling distribution of sample mean $$\displaystyle\overline{{{x}}}$$ is approximately normally distributed with mean $$\displaystyle\mu_{{\overline{{{x}}}}}=\mu$$ and standard deviation $$\displaystyle\sigma_{{\overline{{{x}}}}}={\frac{{\sigma}}{{\sqrt{{{n}}}}}}$$.
Mean of sample mean = $$\displaystyle\mu_{{\overline{{{x}}}}}=\mu={133.5}$$
Standard deviation $$\displaystyle\sigma_{{\overline{{{x}}}}}={\frac{{\sigma}}{{\sqrt{{{n}}}}}}={\frac{{{5.2}}}{{\sqrt{{{230}}}}}}={0.343}$$
Required probability can be obtained as follows:
$$\displaystyle{P}{\left({133.6}{<}\overline{{{x}}}{<}{134.1}\right)}={P}{\left(\overline{{{x}}}{<}{134.1}\right)}-{P}{\left(\overline{{{x}}}{<}{133.6}\right)}$$</span>
$$\displaystyle={P}{\left({\frac{{\overline{{{x}}}-\mu}}{{\sigma}}}{<}{\frac{{{134.1}-{133.5}}}{{{0.343}}}}\right)}-{P}{\left({\frac{{\overline{{{x}}}-\mu}}{{\sigma}}}{<}{\frac{{{133.6}-{133.5}}}{{{0.343}}}}\right)}$$</span>
$$\displaystyle={P}{\left({Z}{<}{1.75}\right)}-{P}{\left({Z}{<}{0.29}\right)}$$</span>
$$\displaystyle={0.95994}−{0.61409}={0.34585}{\left[{u}{\sin{{g}}}{s}{\tan{{d}}}{a}{r}{d}{\left\|{a}\right\|}{l}{t}{a}{b}\le\right]}$$
Therefore, probability that a sample of size $$\displaystyle{n}={230}$$ is randomly selected with a mean between 133.6 and 134.1 is 0.3459

### Relevant Questions

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 single randomly selected value is between 133.6 and 134.1.
$$\displaystyle{P}{\left({133.6}{<}{X}{<}{134.1}\right)}=$$?
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. $$\displaystyle{P}{\left({59.5}{<}\overline{{{X}}}{<}{98.6}\right)}=$$?
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.
$$\displaystyle{P}{\left({98.5}{<}\overline{{{X}}}{<}{100.7}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={221.6}$$ and $$\displaystyle\sigma={44.1}$$. You intend to draw a random sample of $$\displaystyle{s}{i}{z}{e}{n}={42}$$.
Find the probability that a sample of size $$\displaystyle{n}={42}$$ is randomly selected with a mean between 221.6 and 229.1.
$$\displaystyle{P}{\left({221.6}{<}\overline{{{X}}}{<}{229.1}\right)}=$$?
A population of values has a normal distribution with $$\displaystyle\mu={29.3}$$ and $$\displaystyle\sigma={65.1}$$. You intend to draw a random sample of size $$\displaystyle{n}={142}$$.
Find the probability that a sample of size n=142 is randomly selected with a mean between 27.7 and 35.3.
$$\displaystyle{P}{\left({27.7}{<}\overline{{{X}}}{<}{35.3}\right)}=$$?
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 $$\displaystyle\mu={116.3}$$ 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 $$\displaystyle\mu={154.5}$$ and $$\displaystyle\sigma={96.1}$$. You intend to draw a random sample of size $$\displaystyle{n}={134}$$.
Find the probability that a single randomly selected value is greater than 167.
$$\displaystyle{P}{\left({X}{>}{167}\right)}=$$?
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.
$$\displaystyle{P}{\left({198.5}{<}{M}{<}{205.6}\right)}=$$?
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 single randomly selected value is between 59.5 and 98.6. $$\displaystyle{P}{\left({59.5}{<}{X}{<}{98.6}\right)}=$$?