# A population of values has a normal distribution with \mu=120.6 and \sigma=48.5. You intend to draw a random sample of size n=105. Find the probabilit

A population of values has a normal distribution with $\mu =120.6$ and $\sigma =48.5$. You intend to draw a random sample of size $n=105$.
Find the probability that a sample of size $n=105$ is randomly selected with a mean greater than 114.9.
$P\left(M>114.9\right)=$?
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Asma Vang

Step 1
Solution:
Let X be the value.
From the given information, X follows normal distribution with mean $\mu =120.6$ and a standard deviation $\sigma =48.5$. The sample size is 105.
Step 2
The probability that a sample of size $n=105$ is randomly selected with a mean greater than 114.9 is
$P\left(M>114.9\right)=P\left(\frac{M-\mu }{\frac{\sigma }{\sqrt{n}}}>\frac{114.9-120.6}{\frac{48.5}{\sqrt{105}}}\right)$
$=P\left(Z>\frac{\sqrt{105}\left(-5.7\right)}{48.5}\right)$
$=P\left(Z\succ 1.204\right)$
$=1-P\left(Z<-1.204\right)$

Jeffrey Jordon
Jeffrey Jordon