WEBVTT
00:00:01.220 --> 00:00:14.360
Given a normal random variable π such that the probability that π is greater than or equal to π minus ππ and less than or equal to π plus ππ is equal to 0.8558.
00:00:14.430 --> 00:00:16.420
Find the value of π.
00:00:17.420 --> 00:00:30.250
Remember, the graph of s curve representing a normal distribution with a mean π and a standard deviation π is symmetrical about the mean, and the total area under the curve is 100 percent or one.
00:00:31.170 --> 00:00:34.300
Letβs add in what we know about our random variable.
00:00:35.130 --> 00:00:42.100
Now our next step would usually be to find the π value for π plus ππ and π minus ππ.
00:00:42.960 --> 00:00:48.550
This is a way of scaling our data or standardizing it in what becomes a standard normal distribution.
00:00:48.900 --> 00:00:52.700
Once we complete this step, we can then work from a single standard normal table.
00:00:53.540 --> 00:00:57.680
Now we shouldnβt worry too much that we donβt currently know the values of π and π.
00:00:57.910 --> 00:01:02.010
Letβs substitute each other of our values of π into the formula and see what happens.
00:01:02.790 --> 00:01:10.080
For an π value of π plus ππ, the π becomes π plus ππ minus π all over π.
00:01:11.230 --> 00:01:12.930
π minus π is zero.
00:01:12.930 --> 00:01:14.190
So these cancel out.
00:01:14.830 --> 00:01:17.390
And weβre left with ππ all over π.
00:01:18.020 --> 00:01:21.460
The πs then cancel out and weβre left with the π value of π.
00:01:22.220 --> 00:01:26.420
For the next value of π minus ππ, it looks like this.
00:01:27.180 --> 00:01:31.730
Once again, the πs and the πs cancel out and weβre left with a π value of negative π.
00:01:32.570 --> 00:01:40.920
So we now know that the probability that π is greater than or equal to negative π and less than or equal to π is 0.8558.
00:01:41.550 --> 00:01:46.350
We do need to be a bit careful here since we canβt currently look this value up in a standard normal table.
00:01:47.090 --> 00:01:52.400
When we look up a value of π in the table, it tells us the probability that π is less than that value.
00:01:52.740 --> 00:01:56.360
In this case though, we need to find the probability between two values of π.
00:01:56.360 --> 00:02:01.820
At this point then, we instead need to remember that the curve is symmetrical about the mean.
00:02:01.990 --> 00:02:05.630
And since π is a constant, we can see that we have complete symmetry here.
00:02:06.540 --> 00:02:12.030
What we will instead do is subtract the probability that weβve been given from one whole.
00:02:12.280 --> 00:02:16.820
This will tell us the probability that π is greater than π or less than negative π.
00:02:17.340 --> 00:02:22.820
So the area represented by the two pink shaded regions on our curve is 0.1442.
00:02:23.650 --> 00:02:30.100
Halving this, and it tells us the probability that π is greater than π is 0.0721.
00:02:30.660 --> 00:02:32.980
We still canβt look up this value in the table.
00:02:33.430 --> 00:02:40.750
But if we subtract 0.0721 from one, it tells us the probability that π is less than or equal to π.
00:02:41.320 --> 00:02:44.020
Thatβs 0.9279.
00:02:44.800 --> 00:02:48.630
This is the probability weβre going to look for in our standard normal table.
00:02:49.330 --> 00:02:53.330
In fact, a value of 1.46 gives us that probability.
00:02:53.740 --> 00:03:01.140
This therefore means that the probability that π is less than or equal to 1.46 is 0.9279.
00:03:01.760 --> 00:03:05.270
And we have shown that π is equal to 1.46.
00:03:05.840 --> 00:03:14.080
Remember, when we usually work with a normal random variable, we would say the probability that π is less than π for example, rather than less than or equal to.
00:03:14.630 --> 00:03:19.630
However, since normal is a continuous distribution, the difference between these two is minimal.
00:03:20.170 --> 00:03:23.870
And we can therefore use less than or less than or equal to interchangeably.