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Conditions for valid confidence intervals. Conditions for confidence intervals worked examples. Reference: Conditions for inference on a proportion. Practice: Conditions for a z interval for a proportion. Example constructing and interpreting a confidence interval for p. Practice: Calculating a z interval for a proportion.

Interpreting a z interval for a proportion. Determining sample size based on confidence and margin of error. Practice: Sample size and margin of error in a z interval for p. Next lesson. Dating in jupiter florida timeTotal duration Video transcript - [Instructor] We're told Della wants to make a one-sample z interval to estimate what proportion of her community members favor a tax increase for more local school funding.

What is the smallest sample size required to obtain the desired margin of error? So let's just remind ourselves what the confidence interval will look like and what part of it from the margin of error, and then we can think about what is her sample size that she dating cafe erfahrungen analverkehr erfahrungen bmw need.

So she wants to estimate the true population proportion that favor a tax increase. She doesn't know what this is, so she's going to take a sample size of size n, and in fact this question is all about what n does she need in order to have the desired margin of error.

Well whatever sample work takes dating coach werden konjugieren deutschland landkarte, she's going to calculate a sample proportion.

And then the confidence interval that she's going to construct is going to be that sample proportion plus or minus critical value, and this critical value is based on the confidence level. We'll talk about that in a second. And so in this case it would be the square root, it would be the standard error of her sample proportion, which is the sample proportion times one minus the sample proportion, all of that over her sample size.

So the margin of error is this part right over here. It's unpleasant, all right. So how do we figure existential dating vkkk ostbayernbus out?

So we could look at a z-table. So this would be 2. You would look up the percentage that would leave 2. So you would actually look up And that's just something good to know. We could of course look it up on a z-table. So this is 1. And so this is going to be 1.

But what about p hat? We don't know what p hat is until we actually take the sample, but this whole question is, how large of a sample should we take. Well remember we want this stuff right over here that I'm now circling or squaring in this less, less bright color, laughing this blue color.

This is our margin of error. And so what we could do is we could pick a sample proportion, we don't know if that's what it's going to be, that maximizes this right over here. Because if we maximize this, we know that we're essentially figuring out the largest thing that this could end up being, and then we'll be safe.

So the p hat, the maximum p hat, and so if you wanna maximize p hat times one minus p hat, you could do some trial and error here. This is a fairly simple quadratic. It's actually going to be p hat is 0. She didn't even perform the sample yet. She didn't even take the random sample and calculate the sample proportion, but we wanna figure out what n to take, and so to be safe she says okay, well what sample proportion would maximize my margin of error?

And so let me just assume that and then let me calculate n. So let me set up an inequality here. We want 1. So that's our sample proportion. That's one minus our sample proportion. Let me just write this as a decimal, 0. And now we just have to do a little bit of algebra to calculate this. So let's see how we could do this. So this could be rewritten as, we could divide both sides by 1. One over 1. And so this would be equal to, on the left-hand side we'd have the square root of all of this, but that's the same thing as the square root of 0.

This is the same thing as two over So two over times one over 1. Let me scroll down a little bit. This is fancier algebra than we typically do in statistics, or at least in introductory statistics class.

All right so let's see we could take the reciprocal of both sides. We could say the square root of n over 0. Now let's see what's divided by two? That is going to be So this would be And so if we take the reciprocal of both sides, then you're gonna swap the inequality.

So it's gonna be greater than or equal to. Let's see I could multiply both sides of this by 0. Let's see 0. And so there we get the square root of n needs to be greater than or equal to 49, or n needs to be greater than or equal to 49 squared. And what's 49 squared? Well you know 50 squared is 2, so you know it's going to be close to that, so you can already make a pretty good estimate that it's going to be D.

But if you wanna multiply it out we can. Nine times four is 36 plus eight is Four times nine, Four times four is 16 plus three, we have And then you add all of that together, and you indeed do get, so that's 10, and so this is a You do indeed get 2, Now it might turn out that her margin of error when she actually takes the sample of size 2, if her sample proportion is less than 0.

But she just wanted to be no more than that. Another important thing to appreciate is, it just the math all worked out very nicely just now, where I got our n to be actually a whole number.

But if I got 2, So I will leave you there. Sample size and margin of error in a z interval for p. Up Next.