00:01
60 % of us adults work during their summer vacation.
00:05
We take a sample of 500.
00:07
So the probability that each person in the sample works during their summer vacation will be 0 .6.
00:13
And we want some probabilities concerning the sample proportion.
00:18
Now initially this is a binomial distribution.
00:21
We have n independent trials, two outcomes for each person, they do work or they don't, same probability p that each one does.
00:28
And the binomial variable is x, the number from a sample of this size that will meet the criteria.
00:35
We're going to take a normal approximation to the binomial.
00:40
So i have this approximately normal curve.
00:43
It's still a distribution for x, the number of success states.
00:47
Its mean is np, its standard deviation is root np1 -p.
00:53
These are just the mean and standard deviation of the binomial.
00:56
But we're looking at the percentages of the sample that will meet the criteria, not the absolute number that do.
01:03
So to go from the number that work to the proportion that work, we're going to divide by n.
01:10
And i'm going to divide the parameters by n as well.
01:13
So mu becomes mu p hat, which is just p.
01:17
Sigma becomes sigma p hat, called the standard error.
01:20
That is p or minus p over n, all in this square root sign.
01:27
So the parameters of our normal approximation are 0 .6.
01:30
This one's probably not a nice round number.
01:32
0 .6 times 0 .4 divided by 500.
01:36
Is it 500? yeah.
01:38
0 .6 times 0 .4 divided by 500 is 0 .00048.
01:44
And we're square rooting that.
01:55
Ok.
01:59
So we want in part a, the probability that the sample proportion is between 0 .55 and 0 .65.
02:06
So we'll get this area here...