We have a variable x that is normally distributed with:

A. Using the data above we must find the 56th-percentile for the random variable x.
First, we need to find the z-score associated with this percentile. How we do that? We must find the value of z that solves the following equation:
![P(ZThe value of z that solves the equation above cannot be found directly, it is solved by looking at a standard normal distribution table.<p>Based on this, we find that the solution is z = 0.151 because from the normal table we see that:</p>[tex]P(Z<0.151)=0.56]()
Therefore, the percentile we are looking for is computed using the following formula:

B. We must find the proportion of the values for the random variable x between 82 and 89.
In mathematical terms, in this case, we must compute the following probability:

Again, we must obtain the z-scores to solve this. The corresponding z-values needed to be computed are:

Now, because the variable x is a normal distribution, then the variables Zlow and Zup have a normal distribution. Therefore, the probability is computed in the following way:

C. Finally, we must compute the following probability:

The corresponding z-value needed to be computed is:

Again, because x follows a normal distribution, then the variable Zlow has a normal distribution and the probability is computed as:

Summary
The results are:
A. 76.208
B. 0.1507
C. 0.0168