Andy W. answered • 03/22/21
Academic Tutor Specializing in Math Subjects
It seems like we are assuming standard normal here. This will be normally distributed with a mean of 0 and standard deviation of 1, or N(0,1).
For a random variable that is normally distributed, I usually see the probability as an area. For example, P(X < 0) = 0.50. This would mean half of the bell curve is shaded in, and that shaded in portion would be the area.
For this problem, we are already given the area, which is 0.243. The probabilities or the areas will all be found on the Z table, and they will be contingent on the Z score. Since this is standard normal, all we have to do is find the Z score based on the area. In the example I gave earlier, if we look up 0.50 on the Z table, we find that it corresponds with Z = 0. Same deal for the 0.243. To properly read the Z table, we look at the leftmost column first to find -0.7. Then the uppermost row, we see 0.00. We put those together to get -0.70. Therefore, c = -0.70.
There is another way to solve this in case you're not given the negative Z scores. We are given P(Z < c) = 0.243. This value is less than 0.50, which means that the Z score we're looking for is going to be negative. We can take the complement of the probability, so this means we will get the area that is right of the Z score. We end up with 1 - P(Z < c) = P(Z > -c) = 0.757. Additionally, this is equivalent to P(Z < -c) because the bell curve is symmetrical. We find that 0.757 corresponds with Z = 0.70. We can now do P(Z < -c) = 0.757. We find the c such that it fulfills that criteria, in which c = -0.70.
You of course can plug this into a normal distribution calculator, though you might not be given the calculator but rather the Z table during a test. For this problem, you'd plug the 0.243 as the inverse because normally you'd find the area based on the Z score. By plugging it in, we get a Z score of -0.69668, which is very close to the answer we have.
