00:01
So there are five different problems here involving z being a standard normal random variable.
00:06
And in part one, we want to calculate various probabilities.
00:10
The first one being the probability that z is greater than 0 .64.
00:16
The second one being the probability that z is less than negative 2 .11.
00:22
And the third one is the probability that z is in between 0 .34 and 2 .1.
00:33
Now the directions did say to calculate those probabilities using the calculator provided, and i am unsure of which calculator that you have at your disposal.
00:45
So i'm going to do this two different ways.
00:46
I'm going to use the standard normal table, and then i'm going to use the texas instruments graphing calculator.
00:52
So to use the standard normal table, we could draw a bell -shaped curve for each of these scenarios.
01:01
And with that bell -shaped curve, the mean, since we are working in z scores, will go at the center and you will have a z score of zero at the center of each of these curves.
01:21
So now when it comes time to do the first one, 0 .64 would be to the right, and we are looking for the probability of being greater than that.
01:37
And for the second one, negative 2 .11, would be to the left of center.
01:46
And we would be looking for the shaded area to the left.
01:50
And then for the third one, 0 .34 is to the right.
01:59
And 2 .16 is to the right.
02:02
And we're looking for the shaded area in between.
02:06
So to use the standard normal table, we will look for the units place and the 10th place along the left side of the table and the hundreds place across the top.
02:22
So in each case, we're going to be looking for the values that were given.
02:34
And in this bottom one, we're going to be looking for two different values.
02:48
And if you look up the first one, you're going to get an area of 0 .73891.
02:55
If you look up the next one, you're going to get an area of 0 .01743.
03:03
And if you look up the two in the third part, you're going to get areas of 0 .6 -3307, and you'll get .98461.
03:19
Now, what you get out of the table is always representing the area of this bell -shaped curve from that z score and to the left.
03:29
So if to the left is 0 .73891, and we want the area of the area of the same, to the right, we have to utilize the fact that the entire curve, the left plus the right, has to add up to one.
03:43
Therefore, to find the probability that z is greater than 0 .64, we're going to do 1 minus 0 .73891, yielding a probability of 0 .2619.
03:58
With the second one, since the chart gives us to the left, and the left is what we were looking for, that would reflect our probability.
04:11
And with the third one, if to the left of 0 .34 was 0 .637, and to the left of 2 .16 was 0 .9461.
04:31
If i want to know what's in between, i would have to find their difference.
04:42
And it would yield a probability of 0 .3514.
04:48
So now i'm going to show you how to utilize the graphing calculator for each of these.
04:55
And i use a texas instrument brand calculator, and it has a function called the normal cumulative density function.
05:03
And when you use that, you need to provide the information about the lower boundary of your shaded area, the upper boundary of your shaded area, the mean, and the standard deviation.
05:20
So for problem number one, we're going to use normal cdf.
05:29
The lower boundary is the 0 .64.
05:33
There is no upper boundary, so we're going to use a relatively large number.
05:38
And in terms of z scores, 5 is a relatively large number.
05:44
The mean of your standard normal curve is always zero, and your standard deviation is 1.
05:50
So for the next one, we're going to do normal cdf, the lower boundary.
05:59
This time there is no lower boundary, so we're going to use a low number in terms of z scores or negative 5.
06:06
Then the upper boundary, followed by the mean and the standard deviation.
06:16
And then for the third one, we'll use normal cdf, followed by the lower boundary.
06:27
Our shaded area is bounded on the bottom or to left by 0 .34 and on the top by 2 .16, followed by the mean and the standard deviation.
06:39
So we're going to go to our graph and calculator now.
06:42
So i'm going to bring in my graph and calculator.
06:44
And on a texas instrument brand graphing calculator, we can find our normal cumulative density function by using the second button and the vers button and it happens to be number two on my menu.
06:56
Now it might be a different number.
06:59
It's still under that same menu depending on which model calculator or which operating system.
07:05
But either way, you're going to find that normal cdf function.
07:09
So for the first one, we're going to use our lower boundary, followed by our upper boundary, followed by our mean and our standard deviation, and we're going to get a probability of 0 .61086013.
07:32
And you could see now that what we got out of the table is very close to what the calculator has generated.
07:40
For the next one, i'll bring in the calculator again.
07:43
We'll do the normal cdf.
07:46
This time the low boundary is negative 5, followed by the upper boundary negative 2 .11, and then our mean and our standard deviation.
07:55
And this time we get 0 .117, 4288 -913.
08:10
And once again, you can see that the values are close in value.
08:16
And then the third one, we'll bring in that calculator one final time, normal cdf, our lower boundary, followed by our upper boundary, and then the mean and the standard deviation, and we're getting 0 .3515 -41929 -292.
08:37
So it was 0 .3515 -419292.
08:47
And for the final time, you could see our probability using the table and our probability using the calculator are very close in value.
08:57
So let's go to number two.
09:00
And with number two, we know that the probability that z being between negative c and positive c is going to have a value of 0 .9216.
09:20
Now in this one, you were allowed to use either the calculator or this table.
09:24
So we're going to do this one utilizing the table.
09:27
I'm still recommending that you draw that bell shape curve.
09:33
And keep in mind that, that z equals 0 is right at the center.
09:37
So if positive c is to the right, and negative c then would be the same distance, but in the opposite direction to the left.
09:55
And this would be 0 .9 -216 in between...