### survey - How are margins of error related to confidence Intervals? - Cross Validated

The margin of error is usually defined as the "radius" (or half the width) of a confidence interval for a particular statistic from a survey. When a single, global . Suppose a 95% confidence interval for the average amount of weight loss on a diet program {Male weight loss narrative} What is the margin of error? You are confident that the observed difference found in the two samples (plus or minus. Key Words: Confidence interval estimation; Margin of error; Interpretations; appreciation of the difference between a random variable (a function) and its realization The above quotes illustrate the commonly accepted use of frequentist.

As another example, if the true value is 50 people, and the statistic has a confidence interval radius of 5 people, then we might say the margin of error is 5 people. In some cases, the margin of error is not expressed as an "absolute" quantity; rather it is expressed as a "relative" quantity. For example, suppose the true value is 50 people, and the statistic has a confidence interval radius of 5 people.

If we use the "absolute" definition, the margin of error would be 5 people. If we use the "relative" definition, then we express this absolute margin of error as a percent of the true value. Often, however, the distinction is not explicitly made, yet usually is apparent from context. This level is the confidence that a margin of error around the reported percentage would include the "true" percentage.

Along with the confidence level, the sample design for a survey, and in particular its sample sizedetermines the magnitude of the margin of error.

## Confidence intervals and margin of error

A larger sample size produces a smaller margin of error, all else remaining equal. If the exact confidence intervals are used, then the margin of error takes into account both sampling error and non-sampling error. If an approximate confidence interval is used for example, by assuming the distribution is normal and then modeling the confidence interval accordinglythen the margin of error may only take random sampling error into account.

It does not represent other potential sources of error or bias such as a non-representative sample-design, poorly phrased questionspeople lying or refusing to respond, the exclusion of people who could not be contacted, or miscounts and miscalculations.

Concept[ edit ] An example from the U.

### Margin of error - Wikipedia

The size of the sample was 1, Maybe in that one, we got 0. And we already have the tools in statistics to think about this, the distribution of the possible sample proportions we could get. We've talked about it when we thought about sampling distributions.

So you could have the sampling distribution of the sample proportions, of the sample proportions, proportions. And it's going, this distribution's going to be specific to what our sample size is, for n is equal to And so we can describe the possible sample proportions we could get and their likelihoods with this sampling distribution. So let me do that. So it would look something like this. Also, if we make the assumption that the true proportion isn't too close to zero or not too close to one, then we can say that, well, look, this sampling distribution is roughly going to be normal.

So we'll have a normal, this kind of bell curve shape. And we know a lot about the sampling distribution of the sample proportions.

### chapter 21 Flashcards by joy day | Brainscape

We know already, for example, and if this is foreign to you, I encourage you to watch the videos on this on Khan Academy, that the mean of this sampling distribution is going to be the actual population proportion.

And we also know what the standard deviation of this is going to be. So, let me, maybe that's one standard deviation. This is two standard deviations.

That's three standard deviations above the mean. That's one standard deviation, two standard deviations, three standard deviations below the mean. So this distance, let me do this in a different color, this standard deviation right over here, which we denote as the standard deviation of the sample proportions, for this sampling distribution, this is, we've already seen the formula there.

It's the square root of p times one minus p, where p is, once again, our population proportion divided by our sample size.

## Margin of error

That's why it's specific for n equals here. And so in this first scenario, let's just focus on this one right over here, when we took a sample size of n equals and we got the sample proportion of 0. And the reason why I had this uncertainty is we actually don't know what the real population parameter is, what the real population proportion is.

But let me ask you maybe a slightly easier question. What is, what is the probability, probability that our sample proportion of 0. Pause the video, and think about that. Well, that's just saying, look, if I'm gonna take a sample and calculate the sample proportion right over here, what's the probability that I'm within two standard deviations of the mean? Well, that's essentially going to be this area right over here. But if you take this statement, you can actually construct another statement that starts to feel a little bit more, I guess we could say inferential.

Appreciate that these two are equivalent statements. And this is really, really interesting because if we were able to figure out what this value is, well, then we would be able to create what you could call a confidence interval.

Now, you immediately might be seeing a problem here. In order to calculate this, our standard deviation of this distribution, we have to know our population parameter. So pause this video, and think about what we would do instead.

If we don't know what p is here, if we don't know our population proportion, do we have something that we could use as an estimate for our population proportion? Well, yes, we calculated p-hat already. We calculated our sample proportion. And so a new statistic that we could define is the standard error, the standard error of our sample proportions. And we can define that as being equal to, since we don't know the population proportion, we're going to use the sample proportion, p-hat times one minus p-hat, all of that over n.

In this case, of course, n is