00:01
We have been given some sample data.
00:02
The sample size n is 2335.
00:06
Out of these, 725 say they believe in ufos.
00:11
We want a confidence interval for the population proportion.
00:15
The applicable formula is p -hat, point estimate, sample proportion, plus and minus the margin of error, z -root p -hat, 1 minus p -hat, over n.
00:26
We have n.
00:27
P -hat is the proportion of the sample that meets the criteria.
00:32
725 out of 2335.
00:36
Not a nice round number, so i'll just leave it like that.
00:40
Z comes from the level of confidence.
00:44
Now, looking at this, initially we have a binomial experiment.
00:47
N independent trials, two outcomes, they believe in ufos or they don't.
00:51
Same probability p, whatever it is, but they do.
00:54
The binomial variable is x, the number in a sample of this believe.
01:00
We take a normal approximation, divide this by n, turning it into a probability distribution for p -hat.
01:08
Now, our sample proportion is somewhere on this curve.
01:11
We don't know where, but to make our interval, we put it in the centre, form the interval around it to contain 95 % of the sampling distribution.
01:20
That leaves 5 % of the tails, so each tail is 2 .5%.
01:24
Z is the z -score to exclude these tails.
01:28
You might have a table of these.
01:29
If not, use the inverse normal function on software of your choice to get the critical value 1 .960.
01:35
Now, let's put these values into our formula.
01:39
The interval looks like p -hat plus minus margin of error, which i'm going to calculate now.
01:46
So i've got p -hat multiplied by 1 minus itself, divided by 2335, square root, multiplied by 1 .96, which gives me a margin of error 0 .0188.
02:07
Two four decimal places, just so i don't get any rounding errors when i add and subtract.
02:12
If i take that away from the point estimate, i get the lower bound 0 .292.
02:22
If i add it onto the point estimate, i get the upper bound, which is 0 .329...