00:01
When estimating the population proportion, we need to gather some background information.
00:06
And one of the things that we can do to estimate a population proportion is to form what's called a confidence interval for that population proportion.
00:17
Now, there are several steps that will go through to do that.
00:19
But i'm going to run through an example that shows you all of those steps here.
00:23
Now, first of all, let's say that we are sampling a random sample of 1 ,748 of the adult population that do not have a tattoo.
00:37
And we're looking at some characteristic of that sample, and we're seeing that 944 out of our sample of the 1 ,748 meet that criteria.
00:53
Well, how would we find a confidence interval for the population proportion that meet that criteria? well, let's say that we are interested ultimately in finding a 90 % confidence interval for the population proportion, where we've gathered this information from the sample.
01:11
The first thing we want to do is to find a point estimate for the population proportion, and a point estimate for the population proportion is denoted p with a habit.
01:23
Over it and we find it by taking the number in the sample that have that characteristic we're interested in divided by the number in the sample.
01:32
So in this example, our p hat is going to be equal to the 944 in our sample that met the criteria we were focusing our attention on, divided by the 1 ,748 that we sampled.
01:53
So our p -hat in decimal form to three decimal places is going to equal p -hat equal 0 .540.
02:07
And i just did that division and got our value.
02:12
Now, to verify that we can actually go ahead and use the critical values from the standard normal distribution in order to form our confidence interval, we need to check two things.
02:24
The first thing we need to check is if n our sample size times p hat times 1 minus p hat multiplied together does that give us a number that's greater than or equal to 10.
02:36
So here we want to know if that 1 ,748 times the 0 .540, and then times one minus the 0 .540, if you do that multiplication and you carry that out, you actually get about 434, and definitely that is greater than 10.
03:16
That number that is the cutoff that we're comparing it with each time we do this verification.
03:20
So that's yes.
03:22
Next is our sample size n, is this 1 ,748 that we picked as a sample, is that less than or equal to 5 % of the population size.
03:35
Well, 5 % in decimal form is 0 .05.
03:39
And then i want to try to think of what the population size is for this scenario.
03:47
Well, if we're talking about the adults without a tattoo, looking at some statistics of current information, we have that about 180 ,000, 275 ,200 people adults in the u .s.
04:09
Do not have a tattoo.
04:12
So if i multiply 0 .05 times that 180 ,275 ,200 number, we get a number that's like a little over 9 million.
04:25
So 1 ,748 is definitely less than roughly 9 million.
04:32
So again, that's yes.
04:34
So that means that my distribution is such of my population proportion is such that i can use the critical values from the standard normal curve.
04:45
Now remember that critical values for different confidence levels, if i'm allowed to use the standard normal curve for the application i'm doing, is if i have a 90 % confidence level, my z sub alpha over two critical value is 1 .645.
05:03
If i have a 95 % confidence level, they're asking me for my z -sip alpha over 2 is 1 .960.
05:10
And if they're asking me to form a 99 % confidence interval, then my z -sip -alpha over 2 % is 2 .575.
05:19
Now, these i would suggest you commit to memory because they show up over and over and over, and it's easier than just recreating them every single time.
05:27
If you want to find out where those values were found, there's an earlier video.
05:32
Of a problem that shows you exactly how to find those critical values.
05:38
So let's get back to the task at hand...