00:01
For this question, we were told that test scores follow a normal distribution with a mean of 75 and a standard deviation of 10.
00:09
And for part a, we want the probability of x being less than 80.
00:15
So this is the probability of a test score smaller than 80.
00:19
If this graph represents the normal distribution for the test scores, we have a mean of 75 right in the center, 80 would be approximately here, and the probability of x is less than 80, is equal to the area under the curve, and to the left of 80.
00:39
That corresponds to the area of this blue -shaded region.
00:46
Now we can find this area, or this probability using the standard normal table.
00:51
First we must standardize the normal random variable according to this formula.
00:59
And if we do that, we have probability that z is less than half.
01:08
Now that we have expressed as a cumulative probability, we can look up 0 .5 in the standard normal table.
01:15
We find that that corresponds to a cumulative probability of .6915.
01:26
So the probability that x is less than 80 is .6915.
01:31
Then for b we want the probability that x is greater than 60.
01:42
So this time 60 is around here.
01:47
The probability that x is greater than 60 is the area under the curve and to the right of 60.
01:55
And since the total area under any density curve is 1, the area under the curve to the right of 60 is equal to 1 minus the area under the curve to the left of 60.
02:06
That means that we can express this probability as 1 minus, the probability that x is at most 60.
02:16
And if we standardize, according to this formula, we have 1 minus the probability that z is at most minus 1 .5.
02:35
And now let's look up minus 1 .5 in a standard normal table.
02:39
That corresponds to cumulative probability of 0 .0 6x8.
02:51
So this probability comes out to 0 .9332.
03:00
Then for part c, we want the probability that x is between 50 and 85...