00:01
So in this question we have that a set of test scores x are distributed normally with a mean of 120 and a standard deviation of 5.
00:10
And we're going to use the 68, 95, 99 .7 rule.
00:16
So what that means is that the probability that x is between 115 and 125 is 68%.
00:25
The probability that x is between 110 and 130 is 95%, and the probability that x is between 105 and 135 is 99 .7%.
00:41
We also have symmetry that tells us that the probability that x is less than 120 is the probability that x is less than 120 minus, y is equal to the probability that x is greater than 120 plus y.
01:03
So this is our symmetry.
01:06
So then first of all, what's the proportion of scores less than 120? well, the probability x is less than 120 is equal to the probability that x is greater than 120 from symmetry, but this is one minus the probability x is less than 120 by the definition by the law of total probability.
01:24
So twice the probability x is less than 120 is 1.
01:28
So the probability that x is less than 120 is a half or 50%.
01:36
Part b, the percentage of scores greater than 125.
01:39
So the probability x is greater than 125 is 1 minus the probability x is less than 125, which is 1 minus the probability x is less than 125 minus the probability x is less than minus 120, uh, uh, than 115, uh, minus the probability x is less than 115.
02:07
So that's one minus the probability that x is between 115 and 125, minus the probability x is less than 115.
02:16
But remember that symmetry tells us the probability x is less than 115 is the probability x is greater than 125.
02:27
So that means twice the probability x is greater than 125 is 1 minus this probability which we know from our rule is 68%.
02:36
So that's 32%.
02:39
So that means the probability x is greater than 125 is 16%...