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Confidence Intervals

Confidence Intervals

Author: Ryan Backman
Description:

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

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Hi. This tutorial covers confidence intervals. So let's start with a situation here. So suppose you're interested in estimating the mean salary of recent graduates of a local university. If you were presented with the following two interval estimates-- so let's take a look at interval 1 is the interval $35,000 to $39,000. We're talking about dollars here. And interval two is $27,000 to $47,000. So the question is now, which interval is more likely to capture the population parameter?

So if we take a look at these two intervals, we can obviously see that the second interval is much wider than the first interval. So to answer this question, which is more likely to capture the population parameter, we would say the second one because the second one is a wider interval. So it's more likely to capture the population parameter. So we're always going to be more confident in a wider interval. So we would say, in this case, number 2 is the interval that we're more confident in.

So both of those two intervals are what are called confidence intervals. So a confidence interval is an estimate found by using a sample statistic and adding and subtracting an amount corresponding to how confident we are that the interval created captures the parameter. So what I did is I rewrote each of those previous intervals as the statistic plus or minus this kind of secondary amount here.

So the first one was $37,000 plus or minus $2,000. Interval 2 is $37,000 plus or minus $10,000. So in this case, our statistic was $37,000. That would be our estimate. And these two values are what are called margins of error. And that margin of error is the amount corresponding to how confident we are that the interval created captures the parameter. So this had a much larger margin of error, which allows us to be more confident in interval 2.

So if we wanted to be highly confident, say 99%, we would need a bigger interval than if we are somewhat confident, say 90%. So these are called confidence levels, and these are both common confidence levels-- a 90% confidence level and a 99% confidence level. Other common ones are 95%, 98%.

So to interpret that confidence level, a 90% confidence interval means that if we did the exact kind of sample and creating confidence intervals 10 times, then we would expect one of those 10 times our interval would not capture the parameter due to sampling error. So with 90%, if we repeated the process 100 times, 90% of them would produce an interval that captures the parameter. 10 of them would not. And again, that's just due to that sampling error. So this tutorial covered confidence intervals. Thanks for watching.

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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