00:02
It stated that 9 % of college graduates under the age of 26 are unemployed.
00:08
And so based on this, we are asked, what is the probability that at least 200 out of 214 are employed? so if 9 % are unemployed, that means the proportion that are employed is 100 minus 9%, which is 91%, or 0 .91 as a decimal.
00:30
And so we consider a sample of 214 of them.
00:36
The number that are employed.
00:40
And so what is the probability that x is at least 200? so in this situation, each of the 214 sampled can be thought of as bernoulli trials because there are two outcomes of interest, either employed or unemployed.
01:01
And if we can assume that their outcomes are independent, for example if it's a random sample and therefore the probability for each of them being employed is .91, then the random variable x, which is the number out of 214 who are employed, follows a binomial distribution, with parameters n equals 214 trials, and p equals 0 .91 probability success on each trial.
01:30
Now when n, or rather n times p is at least 10, and n times 1 minus p is also at least 10, which is the case here, 91 % of 214 and 9 % of 214 are both greater than 10.
01:56
This means we can use the normal approximation to the binomial distribution.
02:00
That tells us that x is approximately normally distributed with a mean equal to n times p, and a standard deviation of square root of n times p times 1 minus p.
02:21
And so x is approximately normally distributed with the mean of 194 .74, and a standard deviation approximately 4 .186.
02:38
So if we wish to find the probability that x is at least 200, we first re -express it in terms of a cumulative probability using the complement rule is equal to 1 minus probability that x is is less than 200, or 1 minus the probability that x is at most 199.
03:06
Once we have it expressed in terms of a cumulative probability like this, we can apply the continuity correction factor.
03:13
That's because we're modeling a discrete distribution, the binomial distribution, using a continuous distribution, the normal distribution.
03:23
So we apply the continuity correction factor like this...