so before we had talked about the different confidence levels, how we have 90%, 95 and 99% confidence levels.
And we also saw that as our confidence levels increase, so did our margin of ignorance.
Alright. So our Comp. As our confidence levels one from 90 to 95 to 99% As they increase, so did our margin of errors.
So what we were left here to do, as we had to choose that between precision and confidence.
And so this this part make sure you read through this, because we got we got.
We go over that fact that going back and and talking about the difference between the confidence levels, and then what?
That's what that does to our margin of errors.
Okay. So now we're going to focus on
What happens if we increase our sample size
Alright, so ideally. We would like to have both precision and contact when we construct confidence intervals, so can we obtain that if we increase our sample size, alright ideally, we would like to have a narrow interval and when we have a narrow interval that means our margin
of error is smaller, because our interval is smaller. So our margin of error smaller.
So like a 90% confidence level would have the the narrower venturable when you compare it to 95 and 99 so would also have a smaller margin of air, so ideally would like to have a narrower interval and a high level of confidence right
okay, So the 90% gives us the narrow interval with the smaller margin of air.
So if we can help both of them, then that would be ideal.
If so, we can achieve this by increasing the sample size.
So that's what this example does for us It
It just talks about how, if we do increase our sample size, we could see that
We can.
Decrease our margin of error. So that's that's what we see here.
So a larger sample size gives a smaller margin of there.
The larger sample size is, gonna give us a smaller margin of error.
Alright, So I'm just gonna go ahead and write down some information here.
So I'm gonna pause. The video and write down all the information so that you have it.
And I want you to do the same thing. Pause the video, and then write down the formula for confidence interval and try to identify everything here in the problem.
no. I went ahead, and I broke down our confidence. Interval formula.
So you can follow along the question says that national surveys show that 43% of American adult support the legalization of marijuana.
But we really want to find the proportion of students at this community college who support the legalization of marijuana.
So we're interested in estimating the population proportion of students at this community college.
So the 43% here doesn't really factor in any of our calculations, because this is extra information for us.
The population proportion. We want to estimate is the proportion of community college students who support legalization.
Okay. So there's 2 surveys that were conducted.
One was size 100, So I'm gonna call that N one equals 100 and then the second sample was 400 so and 2 would be 400.
Little N. Always represents a random sample, because we have 2.
I call one n one and the other one and 2. Now the results from both of the surveys gave us a sample proportion of 55% who are in favor.
So the P. Hat is point 5 5. Now it happened in both of the samples.
And that it's that surprisingly so we wouldn't.
We don't expect our p-hats to be the same in the different samples.
It just happens to be the same for this particular scenario.
So we have point 5, 5 for the first sample and point 5 5 for the second sample.
It just happened to give us the same P. Hat for both samples, which were different samples.
One was the size 100 and the other one was the size 400, so I make sure that I put P.
Hat one for the first sample, and p hat, 2 for the second sample.
So we're able to know which sample we're talking about.
Now look at our formula. So this part here is P.
Hat. So I just wanna make sure. You see where I'm getting this from.
The is P. Hat. This is our C. And then this right here is our standard error.
Okay. So this part of the formula, with the square root, that's our standard error.
So when we plug everything in the P. Hat, and the one minus p hat, and if you notice, if you do one minus P hat on the side, which is one minus point, 1 5, you can do This on the side so you already know what to plug in for the one minus p so point 5, 5, if you do
one minus point 5 5 you get. You get the point 4, 5 is what I meant to write.
Point, 4, 5,
Okay, And so that's what you see here. The P.
Had times of one minus p hat, and that gets divided by the 100, which is our sample size.
Okay, So because you might be using a different calculator than I am, I want you to go ahead and just pause the video and try to see if you can find what the standard error is.
so on the side. This is what you should have done if you're if you can't put this in your calculator all at once, then I'm just helping you with the calculation on the side here just for the standard error part, and then we're gonna take it and multiply, it
by the 1.9 6. Now we're using 1 point, 9, 6, because this was a 95% confidence.
So remember that we're asked to construct a 95% confidence.
So the Z. That is, representing the 95% confidence is 1.9, 6.
So we're constructing 2 confidence intervals to 95% confidence intervals, one with size, 100.
And the other one was size 400. Okay? So the 1.9, 6 doesn't change because they're both 95% confidence.
The P. Hats do not change, because both samples have. P.
Had a point 5 5. So really, the only thing that's changing is your standard error.
So that's why I'm doing the calculation on the side.
Okay, So it's the square root of p hat times, one minus P hat divided by N.
So if you just do the numerator part point 5 5 times point 4, 5.
It's point 2, 4, 7, 5, Make sure you divide that by 100, and when you do, that's point 0, 0 2, 4, 7, 5.
Now remember, we still have to take the square root of all that right.
So don't forget to take the square root. If you take the square root you're going to end up with a standard error Point 0. 4, 9.
So the question is, how do I? Round this point? 0 4, 9, 7, 4 points are all 4, 9, 7, 5, 9, 3,
7, and a continuous alright. So you want to be careful not to round to, not to round too early.
And also I can't write too many places, either, cause you know they don't really, fit.
But you need to be careful not to round to to fewer decimal places, so points are 497 point, 4, 4, 9, 7, 4, 9.
I'm gonna leave it. Points are 4, 9, 7, 4, 9.
So I can take it to 1, 2, 3, 4, 5, 6 places.
That should be pretty safe. Right? Alright. So we still have to multiply by that 1 point.
9, 6, so you're going to do the 1.9, 6 times our standard error, which is point 0, 4, 9, 7, 4, 9, right?
So this purple, my purple marking here That's the margin affair, That's what this point 0: 9, 8 does. Okay.
But remember, I was very careful for my standard error. I kept it to like 6 decimal places, just to be safe just to make sure I can get that point through a nine-eight margin of error right here.
Alright so now I'm going to do 1.9, 6.
So the writing and purple. It's 1.9 6 times points point 0, 4 9.0, 4, 9, 7, 4, 9 alright.
So when I do this calculation, I get point 0 9, 7, 5.
So now we're okay rounding. Think we're going to round it?
3 to 3 places here, so that 5 Well, it's a 5 or bigger, so that takes that 7.
And it's exactly what we see here. It's points around 9, 8, so it rounds to points around.
I need. I wanted just to take a minute to do this to help you guys through some of these calculations, But of course, if you have a scientific calculator or something where you can just enter it as you see it.
Well, then you'll get points around 9, 8, right away, If you don't, then you're going to have to do the calculation inside the square root first.
So notice how I did. The numerator, and then I did that, and then I put in the denominator, and I do it step by step.
Okay. So you'll notice how we ended up coming up with this margin of error.
So the Z Times. The standard error is the margin of error.
Okay, Now, let's go ahead and pause the video.
And see if you can come up with the standard air. Here, see if you can come up with the value and site the square root here, and then take the square root of whatever the insight is.
So whatever we did here, go ahead, and do again for the bottom, one just for practice.
alright. So the only thing that would have changed in our calculation for our standard error was the 400 right, because right this one had a larger sample size, 400 so let's see what the standard error is you multiply the numerator that's still the same divided by 400
icon. This number my calculator gave me scientific notation.
But that what that really means is that there are 3 zeros in front of the 6, so that Decimal point got moved over to the left 4 times.
Okay, So if you were not sure what that number meant That's what it meant.
Now, remember, you start to take the square root of this to get your standard air.
So my standard error for the second confidence interval, with the larger sample size.
That was the square root of point 0 0, 0, 6 1, 8, 7, 5.
So if you take the square root of that, we got point 0 2, 4, 8, 7 point, 0, 2, 4, 8, 7.
I'm just gonna move that one more place. So I rounded it to 1, 2, 3, 4 to what is that?
6 places again. I'm just I'm just being conservative here, and trying to make sure I don't round too early, because that will change my margin of error.
It will change it So 6 places is pretty safe. Okay? So remember that I saw the multiply by 1 point, 9, 6.
So 1.9, 6, which is the Z. Whenever we have a 95% confidence that gets multiplied to my point, 0, 2, 4, 8, 7, 5.
That was my standard error. Okay. So when you multiply these 2 numbers, so 1.9, 6 times point 0, 2, 4, 8,
7, 5. That gives you your margin of error. Now we're rounding it to 3 places.
So point 0 4, 9 is appropriate. Okay, So this is what I wanted to.
This is what I wanted to make sure that use that you're able to understand.
So as the sample size increases and I'll use red here as your sample size increases.
So we had here. We had 100 sample sizable, 100, and here we had a sample size of 400, so as a sample size increase.
Look what happened to your margin of error. It went from Point 0 9, 8 to point 0, 4, 9.
So, as a sample size increased, the margin of error decreased
Alright. Now we kept the same 95% confidence.
So we were able to reduce our margin of error with this 95% confidence by having a larger sample size
Alright. So the larger sample size really does give a smaller margin of air.
So that's good news. Now, before we knew that larger samples
Before we knew the larger samples do give us better estimates for the population.
Proportion; and we also know that sample size affects variability and sample proportions.
Proportions from larger samples, very less right proportions from larger samples.
Very lost. So proportions from larger samples give us a smaller standard deviation
If there's less variability in a sampling distribution, the standard error is smaller.
So if your standard error is smaller, then your margin of error is smaller, because we use our standard error to find the margin of error Right?
So that's what this says. Since we use the standard error to find the margin of error, larger samples will produce a smaller margin of error.
So, if you're standard error is smaller. Then your margin of error is going to be smaller.
Okay, So that's the main point from this example is that we can reduce our margin of error if we are able to increase our sample size.