 geduiwelh

2021-02-08

A population of values has a normal distribution with $\mu =204.3$ and $\sigma =43$. You intend to draw a random sample of size $n=111$.
Find the probability that a single randomly selected value is less than 191.2.
$P\left(X<191.2\right)=$?
Find the probability that a sample of size $n=111$ is randomly selected with a mean less than 191.2.
$P\left(M<191.2\right)=$? l1koV

Solution for Step 1: Make X a random variable.
Using the information provided, X exhibits a normal distribution with mean $\mu =204.3$ and a standard deviation is $\sigma =43$.
Step 2 Next, determine the likelihood that a single randomly chosen figure will be less than 191.2.
$P\left(X<191.2\right)=P\left(\frac{X-\mu }{\sigma }<\frac{191.2-204.3}{43}\right)$
$=P\left(Z<-0.305\right)$

As a result, 0.3802 percent chance exists that a single randomly chosen figure will be lower than 191.2.
Step 3
A sample size is determined from the provided data $n=111$.
Consequently, the likelihood that a randomly chosen sample has a mean below 191.2
$P\left(M<191.2\right)=P\left(\frac{M-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{191.2-204.3}{\frac{43}{\sqrt{111}}}\right)$
$=P\left(Z<\frac{\sqrt{111}\left(-13.1\right)}{43}\right)$
$=P\left(Z<-3.210\right)$

Therefore, there is a 0.0007 percent chance that a randomly chosen sample has a mean below 191.2.

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