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Confidence Intervals

Confidence Intervals

Author: Katherine Williams
Description:

Identify the z-critical value needed for a given confidence interval.

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Tutorial

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This tutorial covers confidence intervals. Confidence intervals are estimates found by using a sample statistic, then adding and subtracting an amount that corresponds to how confident we are that the interval created captures the parameter. So we're looking at comparing how what we've set up compares to the population. And if we're really confident, based on our interval, like 99%, we're going to need a bigger interval than if we're less confident, only 90%.

By making the interval wider, we can be more certain, more confident that the population parameter is contained inside of it. If we make our interval smaller then we're less confident. We don't have as much room to play with. We're not as confident that the population parameter is inside.

Now, one thing to be careful about is if we're 90% confident, then if we did the same kind of sampling and created confidence intervals 10 times, then we would expect that one of those 10 times would not capture the parameter due to sampling error. I'll show more of what I mean. So here, we're doing a 95% confidence interval. So then it would mean if we tested hundreds times, we would expect that 95% of those tests would capture the parameter.

So here, we have 50 different confidence intervals that we've constructed. And we've done the sampling 50 times. So we're going to expect that 2 of those times, maybe 3 of those times, we would not capture the parameter. And this isn't because our test was done incorrectly. It's just from sampling error.

So if we look across this bar here is showing our confidence interval. And this blue dot is showing the mean for that sample. The red line is showing the population mean. So notice more often than not, the sample mean is not the same as the population mean. And that's going to happen. That's OK.

But, more often than not, the interval that we created around that mean does capture the population. Sometimes our sample mean was low. Sometimes it was high. But in most cases, the bar crosses through the population mean, so that we're seeing are interval does capture the population mean.

But there are some times where it doesn't happen. If we look right here, this confidence interval never crosses the population mean. So this confidence interval does not contain the mean. Similarly, over here, this confidence interval is entirely above the mean. So it does not capture the population mean.

So in both of those cases, there's just some sampling error. It's not that we've done something wrong. It's not that our test is faulty. It's just that due to the fact we're only 95% confident, when we sample 50 times, there's probably going to be about 2 or 3 times where we don't capture the mean. And that's what happened there.

This has been your tutorial about confidence intervals. Other tutorials will go through how to calculate them and how to interpret them.

Z-Tables

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Terms to Know
Confidence Interval

An interval that contains likely values for a parameter. We base our confidence interval on our point estimate, and the width of the interval is affected by confidence level and sample size.

Formulas to Know
Confidence Interval

C I equals P o i n t space E s t i m a t e plus-or-minus M a r g i n space o f space E r r o r

Confidence Interval of Means

C I space equals space x with bar on top space plus-or-minus space z asterisk times space bevelled fraction numerator sigma over denominator square root of n end fraction

Confidence Interval of Proportions

C I space equals space p with hat on top space plus-or-minus space z asterisk times space square root of fraction numerator p q over denominator n end fraction end root