mattgondek4

2021-01-10

A population of values has a normal distribution with mean =136.4 and standard deviation =30.2. A random sample of size $n=158$ is drawn.
Find the probability that a single randomly selected value is greater than 135. Roung your answer to four decimal places.
$P\left(X>135\right)=$?

bahaistag

Step 1
The Z-score of a random variable X is defined as follows:
$Z=\frac{X-\mu }{\sigma }$
Here, $\mu \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\sigma$ are the mean and standard deviation of X, respectively.
Step 2
Consider a random variable X, that defines the variable of interest.
According to the given information, X follows normal distribution with mean ${\mu }_{x}=136.4$ and the standard deviation of ${\sigma }_{x}=30.2$.
The probability that a single randomly selected value is greater than 135 is,
$P\left(X>135\right)=1-P\left(\frac{X-\mu }{\sigma }\le \frac{135-136.4}{30.2}\right)$
$=1-P\left(Z\le -0.046357615\right)$

Therefore, the probability that a single randomly selected value is greater than 135 is 0.5185.

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