Using a nonstandard normal distribution to evaluate aircraft load capacity

Intro

Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to use a nonstandard normal distribution to evaluate aircraft load capacity.  Here's our problem statement: Before every flight, the pilot must verify that the total weight of the load is less than the maximum allowable load for the aircraft.  The aircraft can carry 38 passengers, and a flight has fuel and baggage that allows for a total passenger load of 6,384 pounds.  The pilot sees at the plane is full and all passengers are men.  The aircraft will be overloaded if the mean weight of the passengers is greater than 6,384 pounds divided by 38 (or 168 pounds).  What is the probability that the aircraft is overloaded?  Should the pilot taking any action to correct for an overloaded aircraft?  Assume that weights of men are normally distributed with a mean of 173.1 pounds and a standard deviation of 37.7.

Part 1

OK, the first part of this problem is asking us for the probability that the aircraft is overloaded.  To get the probability, we're going to be using the normal distribution calculator in StatCrunch, because here it says that all the passengers are men and we're assuming that the weights of the men are normally distributed.  So I need to open up StatCrunch.  There's no icon with data for me to put into because there's no data to load into StatCrunch.  So I'm going to come up here to Question Help, and then in the dropdown menu select StatCrunch.  Then I'm going to select this arrow here in the upper right corner so I can move that window out.  And let's resize this so we can see everything a little bit better.

OK, now here in StatCrunch, I go to Stat --> Calculators --> Normal.  Here I'm told that the mean of my distribution is 173.1, so I'm going to put that in here --- 173.1.  And we know the standard deviation is 37.7, but keep in mind when you're calculating probability with the distribution and you have more than one element that you're selecting, then you want to adjust the standard deviation because the standard deviation is a biased estimator.  So whenever you have the probability of more than one, you got to adjust for that bias in your biased estimator.  The way we do that with standard deviation is to divide by the square root of the sample size that we want to take, which in this case here is 38.  So I want to take that standard deviation of 37.7 and I'm going to divide it by the square root of 38.  And this is the number that I want to use for my standard deviation because this number accounts for the bias in my biased estimator.  So in order to avoid any transcription errors, I'm just going to copy and paste that bad boy right in here.  Excellent.

OK, now for this part down here --- let's get the calculator out of the way --- it says we want the probability that the aircraft is overloaded.  It'll be overloaded if the weight --- the mean weight --- is greater than 168.  So this needs to be greater than 168.  I hit Compute!, and there's my probability that we're overloaded.  I'm supposed to round to four decimal places, so I'll do that here.  Excellent!

Part 2

And now the second part of this problem asks, "Should the pilot take any action to correct for an overloaded aircraft?"  Well, look at our probability --- almost 80%.  That's a pretty high probability.  Not an absolute certainty --- I mean, there's a 20% chance that you're going to be OK, but only a 20% chance.  I mean, there's more than 50% chance that you're going to be overloaded.  So I'd say the probability is pretty high, and that means you need to be taking some corrective action.  So we're going to mark yes here.  Good job!

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