yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Worked example finding area under density curves | AP Statistics | Khan Academy


3m read
·Nov 11, 2024

Consider the density curve below. This density curve doesn't look like the ones we typically see that are a little bit curvier, but this is a little easier for us to work with and figure out areas.

They ask us to find the percent of the area under the density curve where x is more than two. So, what area represents when x is more than two? This is when x is equal to two, so they're talking about this area right over here.

We need to figure out the percent of the total area under the curve that this blue area actually represents. So first, let's find the total area under the density curve. The density only has area from x equals one to x equals three.

So, it does amount to finding the area of this larger trapezoid. Let me highlight this trapezoid in red. We want to find the area of this trapezoid right over here, and that should be equal to 1 because all density curves have an area of 1 under the total curve.

Let's first verify that. There are a couple of ways to think about it. We could split it up into two shapes, or you could just use the formula for the area of a trapezoid. In fact, let's use the formula for the area of a trapezoid.

The formula for the area of a trapezoid is you would take the average of this length. We do that in another color. This length and this length, let's see, this is 0.25. 0.25 plus this height, 0.75, divided by 2. So that's the average of those two sides times the base.

The average of this length and this length, let's see, this is 0.25 plus 0.75, which is equal to 1, so the area under the entire density curve is 1, which needs to be true for this to be a density curve.

Now, let's think about what percentage of that area is represented in blue right over here. Well, we could do the same thing. We could say, "All right, this is a trapezoid." We want to take the average of this side and this side and multiply it times the base.

This side is 0.5 high, 0.5 plus 0.75, 0.75 high, and we're going to take the average of that divided by 2 times the base. The base going from 2 to 3 is only equal to 1. So, times 1, and this is going to give us 1.25 over 2.

What is that going to be equal to? Well, that would be the same thing as zero point... What, let's see, 0.625. Did I do that right? Yep. If I multiply 2 times this, I would get 1.25.

So, the percent of the area under the density curve where x is more than 2—this is the decimal expression of it—but if we wanted to write it as a percent, it would be 62.5.

Let's do another example. Consider the density curve below. Alright, we have another one of these somewhat angular density curves. Find the percent of the area under the density curve where x is more than three.

So, we're talking about... see, this is where x is equal to three. x is more than three; we’re talking about this entire area right over here. Well, this is actually somewhat straightforward because if we're saying the area where x is more than three, that's the entire area under the density curve.

The entire area under any density curve needs to be equal to one. Or you could say, "Find the percent of the area under the density curve." Well, the whole density curve is where x is more than three, so one hundred percent. We don't even have to go through the trouble of trying to directly calculate the area.

More Articles

View All
Estimating adding and subtracting 3 digit numbers
[Instructor] What we’re going to do in this video is get some practice estimating adding and subtracting three digit numbers. And so here it says 398 plus 251 is, and this squiggly equal sign means we just need to make an estimate. We wanna know what is…
This Yacht Makes $150,000 Per Week (Here's How)
What’s up guys, it’s Graham here! So this has been the most luxurious week of my entire life, and if you’re curious what a hundred and fifty thousand dollars a week gets you in Croatia, wait no longer! “Graham, welcome onboard Ohana.” “Thank you, I’m ha…
The New Stock Market World Order Has Begun | Recession Warning
What’s up, guys? You here? And, uh, well, this escalated quickly. In the span of one month, mortgage rates have climbed to their highest level in a decade. Morgan Stanley warns that a bear market rally is setting the stage for a correction, with even more…
15 Red Flags Of A Financially Uneducated Person
How do you know if someone is actually rich? There are a lot of people out there who are faking it just for social status or to sell others a lifestyle, and the answer can be summarized in two words: Financial education. Here are 15 red flags of a financi…
TAOISM | The Art of Not Trying
Those who stand on tiptoes do not stand firmly. Those who rush ahead don’t get very far. Those who try to outshine others dim their own light. — Lao Tzu How can we improve when we stop trying to improve? Many people waste their efforts trying to better …
Coral Reefs 101 | National Geographic
(Gentle music) - [Narrator] Coral reefs, their bright, vivid colors can be seen in tropical ocean waters around the globe. Beyond their brilliant appearance lies a hidden significance. Coral are animals. Though they may look like colorful plants, coral a…