Globokim8

2020-10-25

A population of values has a normal distribution with $\mu =239.5$ and $\sigma =32.7$. You intend to draw a random sample of size $n=139$.
Find the probability that a sample of size $n=139$ is randomly selected with a mean greater than 235.9.
$P\left(M>235.9\right)=$?

aprovard

From the provided information,
Mean $\left(\mu \right)=239.5$
Standard deviation $\left(\sigma \right)=32.7$
Sample size $\left(n\right)=139$
Let X be a random variable which represents the value score.
$X\sim N\left(239.5,32.7\right)$
The required probability that a sample of size $n=139$ is randomly selected with a mean greater than 235.9 can be obtained as:
$P\left(M>235.9\right)=P\left(\frac{M-\mu }{\frac{\sigma }{\sqrt{n}}}>\frac{235.9-239.5}{\frac{32.7}{\sqrt{139}}}\right)$
$=P\left(Z\succ 1.298\right)$
$=1-P\left(Z<-1.298\right)$
$=1-0.0971=0.9029$ (Using standard normal table)
Hence, the required probability is 0.9029.

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