non-Standard
Normal Distributions
The normal distribution will be used to calculate very important probabilities (those allowing use to make claims of confidence of an inferential statistic and be able to test a hypothesis.... you know most of science). The first we will see applications for normal probability distributions such as finding the percentage of adult women whose height is between 5 feet 4 inches and 5 feet 7 inches, or finding the probability that a new battery will last longer than 4 years.
Just like in the example of the clock-spinner we will find probabilities by finding the area underneath the curve. Yet as we have already seen the equation is very complicated. So one thing we do is always switch to the standard normal as we just saw.
Find the areas (VIDEO):
1. To the left of any \(z\) value:
Look up the \(z\) value in the table and use the area given.
.jpg "illustration area to left")
2. To the right of any \(z\) value:
Look up the \(z\) value on the table and subtract the area from 1, i.e.
\[1-\text{Area to Left}=\text{Area to Right} \]
.jpg "illustration area to right")
3. Between any two \(z\) value:
Look up both \(z\) values in the table and subtract the corresponding areas, i.e. using the relationship \(z_2\leq z_1\) (just as in the pic below) we have
\[\text{Area to left of }z_1-\text{Area to the left of }z_2=\text{Area between }z_1\text{ and }z_2 \]
.jpg "illustration area between curves")