Applying a nonstandard normal distribution to boat design

Intro

Howdy!  I’m Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we’re going to learn how to apply a nonstandard normal distribution to boat design.  Here’s our problem statement: A boat capsized and sank in a lake.  Based on an assumption of the mean weight of 132 pounds, the boat was rated to carry 50 passengers, so the load limit was 6600 pounds.  After the boat sank, the assumed mean weight for similar boats was changed from 132 pounds to 171 pounds.  Complete Parts A and B below.

Part A

OK, Part A says, “Assume that a similar boat is loaded with 50 passengers, and assume that the weights of people are normally distributed with a mean of 175.2 pounds and a standard deviation of 40.2 pounds.  Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 132 pounds.”  OK, to solve this, I'm going to use the Normal calculator inside StatCrunch, and I know I need the Normal distribution because it says here that the weights are normally distributed.  So inside StatCrunch, I’m going pull up my Normal Calculator by going to Stat –> Calculators –> Normal.

Here's my normal calculator.  I'm going to put in the distribution for the weights that are given here.  So the mean is 75.2, and then the standard deviation — here it says in the problem statement is 40.2, but we've got more than one person on the boat, so therefore we have to do an adjustment to our standard deviation.  And to help with that, I'm going to pull up my calculator here.  So let's see — we've got the standard deviation of 40.2, and that’s divided by the square root of the sample size, divided by 50, the square root — this is the number I need to put in for the standard deviation inside the calculator in StatCrunch.  I’m going to copy that.  So there's my adjusted standard deviation.

Now we want the probability that the 15 passengers have a mean weight greater than 132.  So 132 and then this needs to be “greater than”.  It looks like it's 100%, so the probability is 1.  Well done!

Part B

Now Part B says, “The boat was later rated to carry only 15 passengers, and the low limit was changed to 2565 pounds.  Find the probability that the boat was overloaded because the mean weight of the passengers is greater than 171 (so that their total weight is greater than the maximum capacity of 2565 pounds).  OK, so I got the same distribution here, but I need to adjust my standard deviation.  Now I’ve got 15 passengers instead of the 50, so I pull my calculator back up.  So we’re going to take the standard deviation of 40.2, and this number we’re going to divide by the square root of 15.  I copy that in, and I select the whole number before I copy that in — oh, excuse me, paste it in.  Now I’ve got the right number.  We want the probability that the passengers’ weight is greater than 171, so I need to change this number here to 171.  I hit Compute!, and there's my probability — not an absolute certainty, but not that much of an improvement either.  Fantastic!

Part C

And now, the last part of the problem asks, “Do the new ratings appear to be safe when the boat is loaded with 15 passengers?”  Well, the probability the boat’s going to be overloaded is almost 2/3.  That's a sizable proportion; that’s no small number!  So, yeah, I wouldn’t say the boat — I mean, it’s safer but not appreciably safer.  So yeah, it’s still an unsafe boat.  I wouldn’t want to get on it.

So I’m going to look at my answer options here and select the one that matches that.  “The ratings do not appear to be safe,” but it’s not because of this.  “There is a high probably of overloading.”  I like that one, but let’s check the other two before we hit Check Answer.  No ... no.  Well done!

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