Question 1163131
population mean = 177
population standard deviation is 26


take a sample of 8 people from the population.
probability that the average of the sample is greater than 171 would be calculated as follows:


z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard error which is the standard deviation of the distribution of sample means.


the probability of getting a sample mean less than 171 is found as follows:
z = (x - m) / s becomes z = (171 - 177) / s


s = standard deviation of the population divided by the square root of the sample size.
that makes s = 26/sqrt(8) = 9.192388155 = A


z-score becomes z = (171 - 177) / A = -.6527139519 = B


probability of getting a z-score less than -.6527139519 = .2569702851 = C


what this says is the possibility of getting a random sample of 8 people from the population to have a mean weight of less than 171 is .2569702851.


that means that the probability of getting a random sample of 8 people from the population to have a mean weight greater than 171 is 1 - that = .7430297149 = D.


that's a pretty high probability, so you can conclude that the elevator should not be able to handle 8 people, but something less.


for example, if you want the probability that the elevator to be overloaded is less than 1%, then your one tailed critical z-score is equal to 2.326347877 = E.


that translates to a raw scores as follows:
z = (x - 177) / s becoms E = (x - 177) / A
solve for x to get:
x = E * A + 177 = 198.3846927 = F.


take 1368 and divide it by F to get the maximum number of people = 6.89569332 = G.


if you want to be conservative, then max number of people should be 6.
if you want to be a little less conservative, than maximum of people should be 7.


if you make it 7, then you can calculate the probability of overloading the elevator as follows:


1368 / 7 = a mean of 195.4285714 = H.
z = (x - m) / s becomes z = (H - 177) / A = 2.004764281 = I.


probability of getting a z-score greater than I is equal to .0224940566 = J.


that'a a little more than 2% which should be satisfactory.


bottom line:


maximum of 8 people is not good because probability of overloading the elevator is around 75%.


maximum of 7 people is much better because probability of overloading the elevator is a little more than 2%.


for a visual display, check out the following.


<img src = "http://theo.x10hosting.com/2020/080601.jpg" >


<img src = "http://theo.x10hosting.com/2020/080602.jpg" >


first display assumes maximum of 8 people which makes the maximum mean equal to 171 giving you a probability of about 75% that the elevator will be overloaded.


second display assumes maximum of 7 people which makes the maximum mean equal to 95.4286 giving you a probability of a little more than 2% that the elevator will be overloaded.


2% probability of overloading the elevator is a lot better than 75%.


your solution is that the probability of overloading the elevator when the maximum number of people is 8 = .7430297149 which is roughly 74.3%.


you may have noted the use of capital letters.
those are storage locations in the TI-84 Plus.
they save me from repeatedly entering numbers with large numbers of digits each time.
i just use the value stored in that storage location.


you should also note that the use of the calculator providing the visual display could also have been used to solve the problem without the necessity of finding z-scores.


the only thing that would have been necessary was finding the standard error which, again, is equal to the standard deviation of the population divided by the square root of the sample size in this problem.


this particular z-score calculator can be found at <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">http://davidmlane.com/hyperstat/z_table.html</a>