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
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.
\(\displaystyle{P}{\left({133.6}{<}\overline{{{x}}}{<}{134.1}\right)}=\)</span>?
Write your answers as numbers accurate to 4 decimal places.

Answers (1)

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
0

Relevant Questions

asked 2021-03-18
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.
\(\displaystyle{P}{\left({133.6}{<}{X}{<}{134.1}\right)}=\)?
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 2021-01-05
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 2020-10-26
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.
\(\displaystyle{P}{\left({221.6}{<}\overline{{{X}}}{<}{229.1}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2020-11-30
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.
\(\displaystyle{P}{\left({27.7}{<}\overline{{{X}}}{<}{35.3}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2020-11-09
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.
\(\displaystyle{P}{\left({M}{>}{167}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-02-11
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.
\(\displaystyle{P}{\left(\overline{{{X}}}{>}{117.3}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-01-27
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.
\(\displaystyle{P}{\left({X}{>}{167}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
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.
\(\displaystyle{P}{\left({198.5}{<}{M}{<}{205.6}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2020-12-05
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. \(\displaystyle{P}{\left({59.5}{<}{X}{<}{98.6}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
...