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Confidence Interval for Population Proportion

Confidence Interval for Population Proportion

Author: Ryan Backman
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Calculate a confidence interval for a population proportion.

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Hi. This tutorial covers the confidence interval for population proportion. All right. So let's start with the situation here. So a new telephone survey of likely Minnesota voters finds Barack Obama with 51% support and 46% for Mitt Romney. The survey of 500 likely voters in Minnesota was conducted on October 21, 2012 by Rasmussen Reports. Now, Rasmussen Reports is a public opinion website.

So the margin of sampling error is plus or minus 4.5 percentage points with a 95% level of confidence. So let's see how the article calculated the margin of error of 4.5. Now, the 51% paired with the 4.5% makes up what we call a confidence interval. In this case, it's a confidence interval for a population proportion. Now, what that is is an estimate of the population proportion that uses a sample proportion and a margin of error. OK? So finding a confidence interval for a population proportion is done in a similar way to a confidence interval for a population mean.

All right. So let's investigate the four-step procedure. All right. So step one is you're going to pick a z star that matches the confidence level. So remember the confidence level here was 95%. So what I'm going to do is use a calculator function on my calculator called Inverse Norm to calculate a z star value or a critical z value that matches up with 95%.

So what I'm going to type in is the tail probability of my confidence level. OK? So that ends up being 0.025. My confidence level is 0.95. OK? And that's going to give me a z-score, a z star value of about 1.96. OK? So usually I'm going to just worry about the positive version of that z-score. So z-star equals 1.960. OK, so that is step one.

Step two, I want to check the conditions. So one, the sample observations were obtained randomly. OK? So in this case, we can assume that the phone calls that Rasmussen Reports made were random. OK?

Observations must be independent. If sampling without replacement, n must be less than or equal to 10% of the population. So although we can assume that the observations are probably pretty independent, we did sample without replacement.

So n in this case was 500. And 500 is certainly less than 10% of the population in Minnesota. So I'd say one and two are certainly satisfied.

And condition three, the sampling distribution is approximately normal. OK? So we don't know the actual population parameter, but we can estimate them using the sample statistic. So what I'm going to test is that n times p hat is greater than or equal to 10 and n times q hat is greater than or equal to 10, where q hat is just 1 minus p hat. OK?

So n is 500. p hat in this case was 0.51. So if I multiply those in the calculator, I end up with about 255. 255 is certainly greater than 10. OK?

Now, if I test n times q hat, I'm going to do 500 times 1 minus 0.51. OK? Now, if I do that in the calculator, that's going to give me about 245. And 245 is also certainly bigger than 10. OK? So I would say in this case that my sampling distribution is approximately normal.

All right. step three is I'm going to calculate the interval using the following formula. OK? So the formula I'm going to use to calculate this confidence interval is p hat, my sample proportion. Now, plus or minus my margin of error. So the margin of error consists of your critical z-score, your z star value, times the standard error of the statistic.

Now, generally, our standard error we need to use the assumed population proportion here. But since we're estimating the population proportion, we don't know what it is, what we're going to do instead is replace it with p hat, so our sample statistic. And then instead of q0, I'm going to use q hat.

All right. So let's go ahead and substitute the numbers I know in here. So it's going to be 0.51 plus or minus 1.960 times the square root of p hat is 0.51 again. q hat is 1 minus 0.51. And then I'm going to divide all of that by 500.

OK, so what I'm going to do first is calculate this margin of error. Remember, I was trying to show that the margin of error was about 4.5 percentage points. So let's see if that's correct.

So in my calculator, I'm going to do all of this in one big step. So my z star value times the square root of 0.51 times 1 minus 0.51 divided by 500. And I get a margin of error of about 0.0438. OK? So 0.51 plus or minus 0.0438. OK? So actually, this is pretty close to that 4.5 percentage point. This would end up being about 4.38 percentage points. But we could see that that is pretty close.

And then usually I like to also then just figure out what the boundary points are so I'm going to do 0.51 minus that answer. So that's my lower estimate, 0.466. OK? And then I'm going to call up my margin of error again. But now, I'm going to take 0.51 plus my last answer. And that gives me about 0.554. OK, so 0.554. OK? So this is my confidence interval here for my population proportion.

All right. And then the last thing I want to do is state a conclusion in context. OK, step four. OK? So what I'm going to say then is that I am 95% confident the interval 0.466 to 0.554 captures the true proportion of Minnesota voters who support President Obama.

OK? So that would be a good interpretation of that confidence interval. I'm 95% confident the interval 0.466 to 0.554 captures the true proportion of Minnesota voters who support President Obama. All right. Well, that concludes the tutorial on confidence interval for population proportion. Thanks for watching.

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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