The normal distribution is a very friendly distribution that has a table for finding probabilities and anything else you need. For example, you can find probabilities for
![image0.png](https://www.dummies.com/wp-content/uploads/361017.image0.png)
by converting the
![image1.png](https://www.dummies.com/wp-content/uploads/361018.image1.png)
to a z-value and finding probabilities using the Z-table (see below).
The general conversion formula from
![image2.png](https://www.dummies.com/wp-content/uploads/361019.image2.png)
Substituting the appropriate values of the mean and standard error of
![image3.png](https://www.dummies.com/wp-content/uploads/361020.image3.png)
the conversion formula becomes:
![image4.png](https://www.dummies.com/wp-content/uploads/361021.image4.png)
Don’t forget to divide by the square root of n in the denominator of z. Always divide by the square root of n when the question refers to the average of the x-values.
For example, suppose X is the time it takes a randomly chosen clerical worker in an office to type and send a standard letter of recommendation. Suppose X has a normal distribution, and assume the mean is 10.5 minutes and the standard deviation 3 minutes. You take a random sample of 50 clerical workers and measure their times. What is the chance that their average time is less than 9.5 minutes?This question translates to finding
![image5.png](https://www.dummies.com/wp-content/uploads/361022.image5.png)
As X has a normal distribution to start with, you know
![image6.png](https://www.dummies.com/wp-content/uploads/361023.image6.png)
also has an exact (not approximate) normal distribution. Converting to z, you get:
![image7.png](https://www.dummies.com/wp-content/uploads/361024.image7.png)
So you want P(Z < –2.36).
![z-score table 1](https://www.dummies.com/wp-content/uploads/z-score-table-1.png)
![z-score table 2](https://www.dummies.com/wp-content/uploads/z-score-table-2.png)
Using the above Z-table, you find that P(Z < –2.36)=0.0091. So the probability that a random sample of 50 clerical workers average less than 9.5 minutes to complete this task is 0.91% (very small).
How do you find probabilities for
![image10.png](https://www.dummies.com/wp-content/uploads/361027.image10.png)
if X is not normal, or unknown? As a result of the Central Limit Theorem (CLT), the distribution of X can be non-normal or even unknown and as long as n is large enough, you can still find approximate probabilities for
![image11.png](https://www.dummies.com/wp-content/uploads/361028.image11.png)
using the standard normal (Z-)distribution and the process described above. That is, convert to a z-value and find approximate probabilities using the Z-table.
When you use the CLT to find a probability for
![image12.png](https://www.dummies.com/wp-content/uploads/361029.image12.png)
(that is, when the distribution of X is not normal or is unknown), be sure to say that your answer is an approximation. You also want to say the approximate answer should be close because you’ve got a large enough n to use the CLT. (If n is not large enough for the CLT, you can use the t-distribution in many cases.)