Question 926245
the mean is 61
the standard deviation is 9.
the top 15% of the students will get an A.
you want to know the minimum final exam grade to earn an A.


in the normal distribution curve, getting a score in the top 15% of the class will require a z-score of 1.03643338


15% being greater than a certain score means that 85% are less than that score.


a z-score required for 85% of the scores to be less than that z-score is equal to 1.03643338.


you can find this using a z-score calculator or from the tables.  


the calculator will give you much greater accuracy.


i use the ti-84 plus.


the formula for z-score is z = (x-m) / s


x is the raw score, m is the mean, and s is the standard deviation.


you know z and m and s and you want to solve for x.


from z = (x-m) / s, you can derive x = s * z + m


this becomes:


x = 9 * 1.03643338 + 61 which results in:


x = 70.32790042


students needs a score greater than or equal to 70.32790042 in order to get an A.


this can be visually shown as:


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next question.


you sample 26 exams.


you want to know the probability that 4 of them have a raw score of 50% or less.


the probability of getting a raw score of 50% or less is as follows:


z = (x-m) / s


x = 50, m = 61, s = 9


calculate z-score  to get z = (50-61)/9 which is equal to -11/9.


use your z-score calculator to find that the probability of getting a z-score of -11/9 or less is equal to .1108118649.


your results might vary depending on the calculator you use, but they'll be pretty close to this.


a picture of what this looks like is shown below:


<img src = "http://theo.x10hosting.com/2014/112202.jpg" alt="$$$" </>


the probability of getting a z-score greater than -11/9 is equal to 1 minus the probability of getting a z-score less than -11/9 which makes the probability of getting a z-score greater than -11/9 equal to .8891881351.


you can use the binomial distribution to find the probability that exactly 4 out of the 26 exams will be 50% or less.


the binomial probability formula is p(x) = c(n,x) * p^x * q^(n-x)


x = 4
n = 26
n-x = 22
p = .1108118649
q = .8891881351
c(26,4) = 14950


p(4) = c(26,4) * .1108118649^4 * .8891881351^22 which becomes:


p(4) = .1701557961.


this equals .1702 rounded to 4 decimal places.


a picture of the binomial distribution calculations is shown below:


<img src = "http://theo.x10hosting.com/2014/112203.jpg" alt="$$$" </>