00:01
For this problem, we're going to need to apply the central limit theorem.
00:06
So it says, suppose a category of world -class runners are known to run a marathon in an average of 14 minutes, with a standard deviation of 14 minutes.
00:22
So this is going to be bell -shaped, and the 14 -5 would be in the center, and we would count by 14s to come up with where the first standard deviation is and where the second standard deviation is and so forth.
00:43
But in this particular one, we're going to consider 50 races.
00:48
So now we are going to select n being 50.
00:52
So the central limit theorem states that the average of the averages, or mu x bar, the same thing as mu, so in this case it would be 145, and the standard deviation of the averages, or sigma x bar, would equal sigma divide by the square root of n.
01:23
So in this case, it would be 14, divide by the square root of 50.
01:29
So now let's look at our new curve.
01:32
So this will be our new curve, and the average will be in the center.
01:42
And we'll we are trying to find the 75th percentile of those averages.
01:49
So basically what's happening is we want to know what is that cutoff where 75 percent are below that value.
02:02
So in order to do that, we have to first convert to a z score.
02:06
And to do a z score, we would use our inverse norm feature on our graphing calculator.
02:12
And we will put the area that goes into the left tail.
02:18
The average, well, the average on the z scale or the standard normal scale will be zero and a standard deviation of 1.
02:27
So i'm going to bring in my graph and calculator and we're going to do inverse norm of 0 .75, comma, 0 ,1.
02:39
Oops, sorry about that.
02:41
Let's delete that.
02:45
We want a comma there, 1.
02:47
And in doing so, we are finding that the z score is approximately 0 .67.
02:54
So on the z scale, this cutoff is 0 .67.
02:59
And we're trying to find this right here...