Confidence Interval on a Proportion
D) Confidence interval for the difference of two population proportions of the correct sample size for estimating a population mean or a population proportion. Interpret the confidence interval for a mean or a proportion from a single group. .. proportion is p̂ (called "p-hat"), and it is computed by taking the ratio of the. Construct a confidence interval for the population proportion. 4. Construct a confidence Statistic – z-test vs. t-test: 1. For confidence intervals of the proportion, we use the z-test. n . direct relationship to the size of the population. They are: 1.
And it has some mean, and so the mean of the sampling distribution of the sample mean is actually going to be the same thing as this mean over here-- it's going to be the same mean value-- which is the same thing as our population proportion. We've seen this multiple times. And the sampling distribution's standard deviation, so the standard deviation of the sampling distribution, so we could view that as one standard deviation right over there. So the standard deviation of the sampling distribution, we've seen multiple times, is equal to the standard deviation-- let me do this in a different color-- is equal to the standard deviation of our original population divided by the square root of the number of samples.
So it's divided by Now we do not know this right over here. We do not know the actual standard deviation in our population.
But our best estimate of that, and that's why we call it confident, we're confident that the real mean or the real population proportion, is going to be in this interval. So if this can be estimated it's going to be estimated by the sample standard deviation. So then we can say this is going to be approximately, or if we didn't get a weird, completely skewed sample, it actually might not even be approximately if we just had a really strange sample.
But maybe we should write confident that-- we are confident that the standard deviation of our sampling distribution is going to be around, instead of using this we can use our standard deviation of our sample, our sample standard deviation. That is going to be-- so we have this value right over here, and actually I don't have to round it, divided by the square root of So this is equal to 0. So that's one standard deviation. So it might be from there to there.
So that's what we want. And to figure that out let's look at an actual Z-table. We're going to have to go to 0.
Confidence Interval for a Proportion in One Sample
So this area has to be 0. Now if this is 0. So it's going to be 0. It's going to be 0. Let me make sure I got that right. So let's look at our Z-table. So where do we get 0. So another way to think about it is so this value right here gives us the whole cumulative area up to that, up to our mean. So if you look at the entire distribution like this, this is the mean right over here. This tells us that at 2. So this is 2. If you look at this whole area, this whole area over here, if you look at the Z-table, is going to be 0.
So this whole area right here is So if we look at the area 2. This isn't just 2. So we have to look all the way up into the second to the last column, and we have to add a digit of 8 here. So let me put it this way. There is a it's actually, what, a If you multiply this times 2 you get 0. And we know what this value is right here.
At least we have a decent estimate for this value. We don't know exactly what this is, but our best estimate for this value is this over here. So we could re-write this, so we could say that we are confident because we are really using an estimator to get this value here.
So it is 2. Or you could say that you're confident that the population proportion is within 0. That's the exact same statement. So if we want our confidence interval, our actual number that we got for there, our actual sample mean we got was 0.
So we could replace this, and actually let me do it. I can delete this right here. Let me clear it. I can replace this, because we actually did take a sample. So I can replace this with 0. And just to make it clear we can actually swap these two.
It wouldn't change the meaning. If this is within 0. So let me switch this up a little bit.
Confidence Intervals for a Single Mean or Proportion
So we could put a p is within of-- let me switch this up-- of 0. And now linguistically it sounds a little bit more like a confidence interval.
How do I gauge the precision of an estimated mean or an estimated proportion in a single sample? How do I interpret and calculate a confidence interval for an estimate in a single sample? After successfully completing this unit, the student will be able to: Explain what a confidence interval is.
Interpret the confidence interval for a mean or a proportion from a single group. Use R to compute a confidence interval for the mean in a single group Use R to compute a confidence interval for a proportion in a single group Estimating Population Parameters in a Single Group The goal of exploratory or descriptive studies is not to formally compare groups in order to test for associations between exposures and health outcomes, but to estimate and summarize the characteristics of a particular population of interest.
Typical examples would be a case series of humans who had been diagnosed and treated for bird flu or a cross-sectional study in a community for the purpose of better understanding the current health status and potential challenges for the future. The variables being estimated would logically include both continuous variables e. For both continuous variables e. Recall that sample means and sample proportions are unbiased estimates of the corresponding population parameters. Confidence Intervals For both continuous and dichotomous variables, the confidence interval estimate CI is a range of likely values for the population parameter based on: In practice, however, we select one random sample and generate one confidence interval, which may or may not contain the true mean.
Key Concept A confidence interval does not reflect the variability in the unknown parameter.