Finding Area/Percentage Between Two Z-Scores - YouTube

This video is an example of finding a percentage  of area under the standard normal curve.  

"What percentage of the area under the normal  curve is between the following two z-scores?"  

My first z-score is negative 1.10  and my second z-score is 1.46.  

What you want to do is you want to sketch the area  that they're looking for so I went ahead and drew  

a bell curve or a standard normal curve with  a mean of zero. I put zero in the middle,  

the horizontal axis is a real number line so  a z-score of negative 1.10 will be to the left  

of zero. And then my second z-score is 1.46. Now  because they're asking for a "between" situation,  

I'm going to shade between those two values.  If they asked for the area greater than 1.46  

and less than negative 1.10, I would be  shading those two outer tails instead.  

I'm going to do this two ways. I'm going to use  a standard normal table and then also the TI-84  

calculator. If you don't have a TI-84 calculator,  then an easy way to handle a problem like this  

is to use a standard normal table. To use a  standard normal table, I'm actually going to  

have to use it twice because a standard normal  table reads cumulatively. In other words,  

it's expecting an area from negative infinity up  to a z-score. The table doesn't give an in-between  

case so I'm going to have to deconstruct this bell  curve into two bell curves, one with a larger area  

and then one with the smaller area shaded.  I'm going to subtract out the tail,  

this lower tail that I don't want. Doing that  again I shaded, I drew a bell curve and I shaded  

this larger cumulative area from negative  infinity up to positive 1.46, that's this  

larger area. Then I want to subtract out the  smaller area because I don't want this tail.

Again it's this larger area,  subtracting out the piece we don't want,  

going to the standard normal table for this first  larger area. I expect it to be greater than 0.5  

because I've shaded over half of this  bell curve, half of the area is from  

negative infinity up to zero. It's going to  be half the area plus this little bit more.  

Again I'm not expecting an area less than 0.5  or 50% for this one. In a standard normal table  

the z-scores are on the outside broken into  two pieces. We have the units in the 10th place  

along this first column and the hundredths place  across this first row. I need to intersect those,  

so for this first z-score of positive 1.46, I'm  

going to read down to 1.4 and over  to .06. That first area is 0.9279,  

that's this piece right here. Oops I  think I copied it wrong, one point four

six point nine two seven nine, that's double  checking sorry about that, and then using the  

table again for this negative z- score. I want  to go to the page that indicates z is below zero.  

These are all negative z-scores on the second  page, I want to go down to negative 1.1  

and then over 2.00. Again you can see that across  the top, it's hard to fit this all in the camera,  

okay so that little left tail area that  I'm going to be subtracting out is 0.1357.  

That's how I get point one three five seven here.  

Doing that subtraction, that's not going  to be the answer doing that subtraction,

I get a between area

of .7922, or as a percent that's going  to be 79.22%. That's how you use a table  

a standard normal table. It's a little  bit clunky, but it gets the job done.  

If you have a TI-84 calculator it's  pretty straightforward. A couple of

things to keep in mind. The place  you want to go on a TI-84 calculator  

is this distribution function or menu right  here, it's right above this "vars" key.

Pressing second and the "vars" button,  that'll get me to my distribution menu  

and option two is "normal cdf" or normal  cumulative cumulative distribution function.  

I'm going to select that. It defaults to the  calculator's version of negative infinity.  

Just like how the table theoretically  starts at negative infinity, or the curve  

starts quote unquote at negative infinity, the  calculator approximates that as negative 1 times  

10 to the 99th power. You could type in negative  1 and a bunch of zeros and that would be just fine  

if your calculator doesn't default to that,  okay that would be for finding an area  

from negative infinity up to a z-score. Your  lower bound would would be negative infinity,  

your upper bound would be 1.46. Here your lower  bound would be negative negative infinity and  

your upper bound would be negative  1.10. With the calculator, however,  

you can do an "in-between" case just by  putting in your lower bound and upper  

bound. We have a lower bound of negative 1.10,  upper bound of negative 1.46. Putting that in,

and you want to leave these two numbers  

as is if you're working with z-scores or  a standard normal situation, the mean is 0  

and the standard deviation is 1 for the standard  normal distribution. Then I'm going to arrow down,

enter, and enter again and rounding to four  decimal places, you'll see that I get 0.7922,  

which is the same value that I got  when I used the standard normal table.