00:01
Once again, welcome to a new problem.
00:04
When we think about the standard normal distribution, when we think about the standard normal distribution, what tends to happen is that we can always apply the empirical rule.
00:23
And with the empirical role, what happens is if we have a mean of a sampling distribution, of sample proportion and a standard deviation of a sampling distribution of sample proportion, then we can see that within one standard deviation of the mean on both sides of the mean, within one standard deviation on both sides of the mean, we have 68 % of the data.
00:58
So within one standard deviation of the mean, we have 68 % of the data.
01:10
And then within two standard deviations of the mean, we have 95 % of the data.
01:22
So this is two standard deviations on both sides of the mean.
01:29
And then, of course, within three states.
01:32
Standard deviations on both sides of the mean.
01:41
We have 99 .7 % of the data.
01:54
99 .7 % of the data.
01:59
And so we have a new problem and we're looking at this problem involving outsourcing.
02:06
So there's a bunch of us companies that have an interest in building up their profit profile.
02:13
And so they use outsourcing to cut their costs.
02:19
There's a magazine that hypothesizes that 54 % of companies have outsourced in the past 2 to 3 years.
02:31
So there's a sample of 555 or 555 that's contacted.
02:38
And we want to determine the probability that the 338 or more companies outsource some or part of their manufacturing.
02:51
So we want to see that.
02:53
So in part a, first of all, we want to get the mean and the standard deviation of the sampling distribution of sample proportions.
03:05
So the mean is the same as the population hypothesized population mean, which is 54%.
03:13
And then, of course, the standard deviation or standard error is the same as radical p, 1 minus p, all over n.
03:27
We do have a sample of 555.
03:31
And so we're going to use that sample to determine the standard error, which is 555.
03:39
That's the sample we're using.
03:42
And the requirement for this problem is to look at the probability of 338.
03:51
So we have to get the p hat.
03:53
The p hat is x over n, which is 338 over the sample size.
04:00
And so the result we get for 338 divided by the sample size becomes the same as 0 .609.
04:12
So this is where we have it.
04:15
0 .609, that's our p .hat.
04:19
And the probability we're looking for is on the higher side.
04:25
So then probability is that p hat is equal to 0 .3 .3.
04:33
609 or more.
04:37
So then we have to get the z score of this particular problem, which is p -hat minus -p, all over radical p1 minus p over n.
04:49
And so the z score helps us determine the probability.
04:54
P -hat is 0 .609 and p is 0 .54.
05:01
We're dividing that by the standard error, which is 0 .54.
05:07
1 minus 0 .54 all over 555.
05:12
And this is helpful in getting the required z score to help us determine the probability.
05:20
So we have to run the math for this problem.
05:24
And the math will become divided by radical .54 times .46 divided by 555.
05:39
And so we get 3 .2619.
05:48
And the probability result for this one, 3 .2619, or just 3 .26 will give us a probability equivalent to 1 minus 0994 .9.
06:18
And that probability will be the same as 0, as really small 0 .1, 2, 3, and 4.
06:32
So it's a really small probability.
06:36
The second part of the problem, if you go back up, is, so we did 338 or more...