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In this video we are going to show you how
- in 5 easy steps – we can use GeoGebra to visualize and calculate values from the Normal Distribution Table.
Typically we use the Normal Distribution Table when working on sampling distributions, confidence intervals and hypothesis testing.
We can also use GeoGebra commands, but this is more fun and easy.
The Normal Distribution Table is based on the normalized Normal Curve, also known as the bell curve. So let's draw this.
Go to the Input Bar and start typing Normal. Since we want the normalized Normal Distribution, we want the mean = 0 and deviation=1. Hit enter.
We get the formal definition of the normal distribution and a rather squashed bell curve. Let's stretch it and then center it.
Put your mouse cursor on the y-axis and press and hold Ctrl (on a PC) and then click and drag the axis up.
Center the graphics view (I click and hold my mouse scroll button and then drag the view).
Now we want to be sure that the area under the whole bell curve is 1. That is the meaning of a "normalized" distribution, the area is 1.
How to we calculate the area? We start all the way over to the left at x=minus infinity and going to x=plus infinity and calculate this area.
But typically a Normal Distribution Table uses 4 decimal places for p and 2 decimal places for z.
So lets go and make our rounding 4 decimal places. Options -> Rounding -> 4 decimal places.
With 4 decimal places, it is "good enough" to start at x=-6 and go to x=6.
How do we calculate area under a curve. We use what is called the integral. So let's make sure the area is 1. Go to the input bar and
start typing Integral. Look for option with function, start value, end value and click on it. Type in f, hit tab, type in -6, hit tab, type in 6 and hit enter.
And that should shade the curve and give us 1. Sure enough. We get a=1 at left and the area under the curve is shaded. Good!
OK – now the Normal Distribution table has values in it. How do we get those values?
1. Create a slider for z. We start it at 0 (the mean) and end at 3 and make our increment 0.01 (matching the 2 decimal places in the table).
2. Find the area under the curve from x=z to x=6 using the integral command again. In the input bar, start typing Integral. Find the same option with
function, start value, end value and click on it. Now, type in f, hit tab, type in z, hit tab, type in 6 and hit enter. The "right tail" under the curve is
shaded and its value is given. For z=1, this value is: b=0.15865. So 16% of the curve is under the right tail.
3. However the values in the Normal Distribution Table is the AREA TO THE LEFT of the TAIL. How do we calculate this?
It is 1-b. So we define p=1-b. Done with construction! Let's check our table.
For z=1, we should see p=0.8413. Good. Let's try another. Set z=1.65. How do we do this precisely. We click once on the slider button
WITHOUT MOVING left or right. The button should "glow". Now we can use our arrow keys to get to 1.65. We see: 0.9505. Check our table.
Good. Now let's go backwards. Suppose we want to find z for p=0.9772. Using our arrow keys we move z until p=0.9772. Look at z. It is 2.
So we can use this little applet to visualize and calculate the Normal Distribution Table.
And now I have a challenger question for you that was a little confusing for us so I thought I would ask you the question.
We know that the rule for the bell curve is 68-95-99 or more precisely 68.3-95.44-99.7
Here z is the number of standard deviations.
When we set z=2, we get p=0.9772 or 97.72%. Why isn't p=0.9544?
If you need some help, there is a hint at the end of the video description.
Have fun!