Two Proportions | STAT /
You can find the confidence interval (CI) for a population proportion to show the statistical probability that a characteristic is likely to occur within the population. Applying the general formula for a confidence interval, the confidence interval for a proportion, π, is: p ± z σp where p is the proportion in the sample, z depends. One can compute confidence intervals all types of estimates, but this short Interpret the confidence interval for a mean or a proportion from a single .. The sample proportion is p̂ (called "p-hat"), and it is computed by taking the ratio of the.
So in our survey, so we had sampled, and we got said that it is good, and we'll say that this is a 1. So we got 1's, or we sampled 1, times from this distribution. And then the rest of the time, so what's left over? There's another who said that it's not good. So said not good, or you could view them as you were sampling a 0, right? So what is our sample mean here? We have 1 timesplus 0 times divided by our total number of samples, divided by It is equal to over You could even view this as the sample proportion of teachers who thought that the computers were a good teaching tool.
Now let me get a calculator out to calculate this. So we have divided by is equal to 0. So our sample proportion is 0.
Now let's also figure out our sample variance because we can use it later for building our confidence interval. Our sample variance here-- so let me draw a sample variance-- we're going to take the weighted sum of the square differences from the mean and divide by this minus 1. So we can get the best estimator of the true variance.
So it's 1 times-- no, it's the other way actually around-- we have samples that were 1 minus 0. Plus the other times we got a 0, so we were 0 minus 0. And then we are going to divide that by the total number of samples minus 1.
That minus 1 is our adjuster so that we don't underestimate. So minus 1. Let's get our calculator out again. And so we have we put a parentheses around everything-- I have times 1 minus 0. So our sample variance is-- well, I'll just say 0. It is equal to-- it is our sample variance-- I'll write it over here-- our sample variance is equal to 0. If you were to take the square root of that our actual sample standard deviation is going to be, let's take the square root of that answer right over there, and we get 0.
I'll just round that up to 0.Confidence Level and Margin of Error
So that is our sample standard deviation. Now this interval, let's think of it this way, we are sampling from some sampling distribution of the sample mean. So it looks like this over here, it looks that over there. 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.
Confidence Interval for the Difference of Two Population Proportions
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. 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. 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. Rather, it reflects the amount of random error in the sample and provides a range of values that are likely to include the unknown parameter.
Another way of thinking about a confidence interval is that it is the range of likely values of the parameter with a specified level of confidence which is similar to a probability.
The Central Limit Theorem states that, for large samples, the distribution of the sample means is approximately normally distributed with a mean: There is often confusion regarding standard deviations and standard errors. Standard deviations describe variability in a measure among experimental units e.
Standard errors represent variability in estimates of a mean or proportion; i. Another way to think of this is that standard deviations describe the variability in a population while standard errors represent variability in the sampling means or proportions.