00:01
In a certain school district in a large metropolitan area, the sat scores over the past five years were normally distributed.
00:10
So that means we could draw the bell -shaped curve.
00:20
It goes on to say that the mean was 1 ,475, so we are going to place our mean at the center.
00:34
You're also been informed that quartile 1 is 1 ,000.
00:41
266.
00:46
Well, the meaning of quartile 1 means that there are 25 % of the data lower than that value, and 75 % of the data is higher than that value.
01:01
So that means 1266 is going to be over here, which is smaller than the mean, and the area to the left, or less than that, is 20266 is 25%.
01:18
So we could say 0 .25.
01:21
And the first thing we want to do here is find the z score associated with the 25th percentile.
01:30
So we want to find this z score.
01:33
And the most efficient way to find z scores when you are provided an area is to use a graph and calculator.
01:42
And on the texas instruments graphing calculator, we have a function called inverse norm.
01:49
And when we use that inverse norm function, we need to provide certain parameters.
01:55
We need to provide the area to the left, followed by the mean, and the standard deviation.
02:03
So to find this z score, the area to the left is going to be that 25%, or 0 .25.
02:11
And because we are working in z scores, the mean of z scores is equal to zero, and our standard deviation is one.
02:25
So i'm going to bring in my graphing calculator, and my graphing calculator is a texas instruments brand calculator, and it happens to be in 83, but all their calculators have the same set of keystrokes.
02:39
So therefore, we're going to hit the second button.
02:43
And the variables button, which gets us into the statistical distributions, and it happens to be number three in my sub -menu.
02:51
Now, it might be another number in the sub -menu based on the operating system or the model calculator.
02:59
So we're going to use inverse norm.
03:01
We're going to put in the area that's to the left, followed by the mean and the standard deviation.
03:08
And we are getting a z score carried out to as many decimal plus.
03:12
Places as possible is negative 0 .67448 -97502.
03:23
Now it does say to enter your answer rounded to three decimal places.
03:30
So when we answer this z score, we're going to say it was negative .674.
03:39
But when we go and use that information for future parts of this problem, we're going to use the entire z score with all of its decimals.
03:52
So the next thing we want to do is we want to use what we just found to help us find the standard deviation.
04:00
So to find the standard deviation, we are going to start with our z score formula.
04:06
And our z score formula says z equals x minus mu divided by sigma.
04:15
And if i were to start by cross -multiplying and trying to isolate sigma, i would get that z times sigma is equal to x minus mu.
04:30
So when it comes time to find what sigma is, i will divide both sides by z.
04:36
And i have a new formula that sigma will equal x minus mu over z...