 # For the standard normal distribution, find the following probabilities. (с) Pr(zleq 2.5) York 2020-11-26 Answered
For the standard normal distribution, find the following probabilities.
(с) $Pr\left(z\le 2.5\right)$
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The z score for a normally distribution of x is,
$z=\frac{x-\mu }{\sigma }$
Where, $\mu$ is the mean and $\sigma$ is the standard deviation.
Calculation:
Consider the standard probability distribution, $P\left(z>2.5\right)$.
Now, the normal distribution is symmetric about mean 0.5.
So this implies,
$P\left(z\le 25\right)=0.5+Pr\left(z<25\right)$
The probability that the z lies between 0 and 2.5 is equal to the area that lies under the curve from 0 and 2.5.
To find the probability look in the column headed by z for the value of 2.5 in appendix C.
In the column headed by A across 2.5 the corresponding value is 0.4938.
Thus the probability $P\left(0\le z\le 2.5\right)$ is ${A}_{2.5}$.
So the value of ${A}_{2.5}$ from the appendix C is 0.4938.
The value of $P\left(z\le 2.5\right)$ is:
$P\left(z<25\right)=05+0.4938=0.9938$
Hence the value of $P\left(z\le 2.5\right)$ is 0.9938.