Help with statistics

Question

An elevator has a placard stating that the maximum capacity is

1408 lb —8 passengers.​ So, 8 adult male passengers can have a mean weight of up to

1408/8=176 pounds. If the elevator is loaded with 8 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 176 lb.​ (Assume that weights of males are normally distributed with a mean of 179 lb and a standard deviation of 34 lb​.)

Does this elevator appear to be​ safe?

The probability the elevator is overloaded is

Answer 1

For this problem, you will need to either calculate a z-score and locate the probability using the Standard Normal Table, or use the Normalcdf function on your calculator. I will explain both methods just in case:

1) z-score method

First, you will need to calculate the standard error, because we are using a sample (where n=8). The formula for standard error is standard deviation/square root of n. So, here, it will be 34/sq.rt. 8 = 12.02.

Now, you can calculate the z-score for the elevator being overloaded (that is, when the mean weight is above 176 pounds). The z-score formula is x-mean/standard error. So, here, it will be 176 - 179 / 12.02 = -0.25.

Knowing the z-score, you can now go to the Standard Normal Table, locate -0.25 as the z-score, and get a value of .4013 in the tail... but we don't want the proportion in the tail, we want what is above our z-score, so we do 1 - .4013, which is .5987

2) Calculator method

You still need the standard error for this method, so you would begin by finding that, as described above. The standard error in this case is 12.02.

Next, go to the "Normalcdf" function on your calculator. On the Ti-84 plus, it is located under 2nd vars. For the lower limit, you'll put in 176, and for the upper limit, just put in a really large number, like 10000000. For the mean, put in 179, and for the standard deviation, put in the standard error (not the original standard deviation because that does not take into account the sample of 8). Enter all of that in, and you should get .5985.

Either way, the answer is roughly 59.9 percent, or .5999.