00:01
Here we're told that we've got a survey of 2 ,272 adults, and of these adults surveyed, 733 of them say they believe in ufos.
00:16
And using this information, we want to construct a 95 % confidence interval for the true population proportion of adults who believe in ufos.
00:25
Let's start by finding our point estimate for the population proportion, p -hat.
00:31
That's simply given by x over n, the total number of people who believe in ufos from our sample, over the total number of people sampled.
00:42
Let's see what this works out to as a decimal.
00:49
It looks like this is about 0 .323, though i will use the exact answer in my calculator for further computations.
00:58
Now, let's recall how we construct the confidence interval, now that we have p -hat.
01:03
You take p -hat, and then you add or subtract the margin of error to get the endpoints of your confidence interval.
01:10
And that margin of error term has a z -score for the level of confidence you want, times the square root of p -hat times 1 minus p -hat over n.
01:21
And as i said, this is the margin of error.
01:24
So the only thing we need to compute this is this z -score.
01:28
So let me explain how we get that.
01:31
Imagine you've got a standard normal distribution, mean 0, standard deviation 1, then this z -score is such that the area under the curve between minus z sub c and plus z sub c is equal to 95 percent of the area under the curve, since that's the level of confidence we're working at.
01:50
Means this area in red here is 0 .95, as the total area under the curve is 1.
01:56
And now, due to the symmetry of the curve, the remaining 0 .05 area is equally distributed in these two tails, which means 0 .025 in each tail.
02:06
And now that lets you write down the area to the left of z -score, 0 .975.
02:15
And now you can use various tools to turn this cumulative probability for the z -score into the z -score itself.
02:22
For example, z -score tables or the inverse normal function.
02:27
I'm going to go ahead and use the inverse normal function on my ti -84, and i'm getting a z -score of 1 .968.
02:36
0 to three decimal places.
02:39
So there's my z -score.
02:41
And now i'm in a position to calculate this margin of error.
02:45
I'm simply going to plug everything we've got into my calculator...