I will be explaining how to solve problem #13.
Question: A professor has found that the grades on the Statistics Final are normally distributed with a mean of 68 and a standard deviation of 15. If only the best 14% of the grade in the class will receive an A, what grade must a student obtain in order to get an A?
First what we want to do is write down the information that we are given:
Standard Deviation: 15
Since it is a normal distribution problem, we know that the mean (68) should be placed in the middle of the bell curve that I encourage should be drawn in order to get a better picture of what the problem is asking for. We are trying to figure out what grade should the best 14 % of students get in order to receive an A. That is another way of saying the top 14% and so, we should be shading a small area on the far right side of the mean (68) on the bell curve.
After having this drawing, we now know that we have to find the z value that corresponds to the right side of this curve. This is how we will proceed into finding this z-value:
We have to convert 14% into a decimal= .1400
Next, since we are looking for the right side value, we do the following:
After getting .8600, we look for the two numbers close to it in the given tables. The two numbers closest to .8600 are .8599 and .8621. From these two numbers, now we have to see which one is the closest to .8600.
.8599 is only one number away and so we choose this one to work with. From here, we look at the z-score that this area falls into, and that would be 1.08.
From here, in order to find the x value that would give us our final answer, we must use the formula x= mean + (z-score)(standard deviation).
Once we plug in the numbers, it would look like this:
x= 68 + (1.08)(15)
Now let’s solve for x, by multiplying the two numbers in the parenthesis first and then proceeding with the addition.
When solving the equation properly, x should equal 84.2
Therefore, a student must obtain at least an 84.2 in order to get an A.