Using a nonstandard normal distribution to evaluate pregnancy and premature birth data

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 pregnancy and premature birth data.  Here's our problem statement: The lengths of pregnancies are normally distributed with a mean of 266 days and a standard deviation of 15 days.  Part A: Find the probability of a pregnancy lasting 309 days or longer.  Part B: If the length of pregnancy is in the lowest 2%, then the baby is premature.  Find the length that separates premature babies from those who are not premature.

Part A

OK, Part A is asking us to find the probability that a pregnancy will last 309 days or longer.  To do this, I'm going to use the normal distribution calculator in StatCrunch, and I know I need the normal distribution calculator because here in the problem statement it says my data is normally distributed.  So the first thing I need to do is open up StatCrunch.  And let's resize this window so we can see everything a little bit better.

Now in StatCrunch, I go to Stat --> Calculators --> Normal.  Here in my Normal calculator, I need to establish the mean and the standard deviation, because the defaults here are for the standard Normal distribution and we have a nonstandard Normal distribution.  Here in the problem statement, it says the mean is 266 days.  So I need to put that in here.  And the standard deviation is 15.  And then I'm asked for the probability that a pregnancy will last 309 days or longer.  Well, look at how this is ordered here. Probability is P, x is my random variable --- that's going to be the 309 days --- but it needs to be 309 days or longer, which means that this is greater than or equal to 309.  So I got to flip that around.  And notice here on the other side of the equals sign is my probability.  This is a probability in decimal form.  I'm asked to round to four decimal places, so I just type that in here.  Fantastic!

Part B

Now, Part B asks for the number of days in which a birth would be considered premature.  So babies who are born on or before how many days are considered premature?  And to do that, I go back to my distribution calculator.  And this is the number we're looking for now.  So we get rid of that premature birth from the problem statement.  We're looking for the lowest 2% of the distribution.  That's the left tail of the distribution.  So I need to change this to 2%.  And now we just switch this around so I can get the left side and not the right side of my distribution.  Now we're at the left tail of the distribution.  This is the lowest 2%, and this boundary here for the tail is the value we're looking for.  Rounding to the nearest integer gives me 235.  Nice work!

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