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Confidence Interval for Population Proportion

Confidence Interval for Population Proportion

Author: Katherine Williams
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Calculate a confidence interval for a population proportion.

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This tutorial talks about confidence intervals for population proportions. A confidence interval for population proportions is very similar to a confidence interval for a mean. There's a couple of key differences.

Now a confidence interval, in general, is an estimate that's found by using a sample statistic, in this case, a sample proportion. And then you add or subtract an amount that corresponds to how confident we are that the interval created captures the population parameter. So in this case, the population proportion. When we're doing these confidence intervals we're going to use p hat and q hat in the formulas because we don't have an assumed population proportion. So this is all based, for our confidence intervals, on our sample proportions.

So our steps are first to pick a z star level that matches the confidence interval. So for 95%, there's a certain level. For 99%, there's a certain level. For 90%, there's a certain level. That's just a number that you can obtain from a chart.

Then you need to check the conditions. Conditions are pretty typical, that the observations come from a simple random sample, that they're independent, and that it's approximately normal. Now the part that's a little bit different is how we calculate our confidence interval. And in this case, we're using the sample proportion. We're adding and subtracting the z star value from that confidence level, and multiplying by the standard error, which in this case, is the square root of the sample proportion that you're successful, times the sample proportion that your failure at, so 1 minus the successful, divided by the sample size. Finally, you're going to state the conclusion in context.

So here's an example. It says, a survey of 59 students finds that 25.4% of the college students vote. And they want us to construct a 95% CI. CI stands for confidence interval.

So first, we need to pick a z star that matches the confidence level. And that z star for the 95% confidence level is 1.96. So for 95%, the z star is always 1.96. I checked the conditions, they're good. Now we need to actually do the calculations. And I'm going to scroll up just a little bit.

So here, p hat. I'm going to start by writing out what my things are. So p hat is 25.4%, so 0.254. The z star is 1.96. The n, we took a sample of 59 students. And then q is going to be 1 minus 0.254, which is the same as 0.746.

So now that we've picked out all the key parts of our problem, we're going to establish the setup and then do the calculations. So we start with p hat. So 0.254 plus, minus the z star, 1.96, times the square root, p hat, so 0.254, q hat, 0.746. And then divided by n, the sample size, 59. I'm just going to make that a little bigger so we know we're taking the square root of everything.

So once we type this into the calculator, and when I calculate this part here, we get 0.254 plus minus 0.1111. 1 Now at this point, you have a confidence interval. You know you're going to add this with this piece and subtract this piece from this piece. But if we actually did those calculations out, we would find we have 0.143 to 0.365. And if we wanted to look that in terms of percents, it would be 14.3% to 36.5%.

So based on this, when we've taken a sample of 59 students and find that 25.4% of them vote, we can be 95% sure that the population parameter, so the percent of college students that vote, is somewhere between 14.3% and 36.5%. This has been your tutorial on confidence intervals for population proportions.

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Terms to Know
Confidence Interval for a Population Proportion

A confidence interval that gives a likely range for the value of a population proportion. It is the sample proportion, plus and minus the margin of error from the normal distribution.

Formulas to Know
Confidence Interval of Population Proportion

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 with hat on top q with hat on top over denominator n end fraction end root