Finding "significant values" using a normal distribution

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find “significant values” using a normal distribution.  Here's our problem statement: Suppose that the sitting back-to-knee length for a group of adults has a normal distribution for the population mean of 24.1 inches and a standard deviation for the population of 1.2 inches.  These data are often used in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats.  Instead of using 5% for identifying “significant values,” use the criteria that a value x is significantly high if the probability of x or greater is less than 1% and a value is significantly low if the probability of x or less is less than one 1%.  Find the back-to-knee length separating “significant values” from those that are “not significant.”  Using these criteria, is a back-to-knee length of 26.5 inches significant?

Part 1

Here we have the first part of our problem, which says, “Find the back-to-knee length separating significant values from those that are not significant.”  To help us with this, we're going to use StatCrunch.  I could alternatively go old school and use the z-tables to find my answer, but that would require me to convert between random variables and z-scores.  I'm going to use StatCrunch because it's much simpler; it does all the conversion for you here inside StatCrunch.

I'm going to go to Stat –> Calculators –> Normal.  Notice the default settings for the Normal calculator are those for the standard normal distribution.  The normal distribution that we have here in our problem statement is not a standard normal distribution, because we have a mean value that's not zero and a standard deviation that is not one.  What I'm going to do is put those population parameters here into these fields in StatCrunch.  This will allow StatCrunch to adjust everything for me and do the conversions that I need for me.

Now the next step is to identify the criteria that we're using for separating out “significant” and “not significant” values.  Notice this is statistician talk.  What we're talking about are values that are usual that we would expect to see, and then there's values outside that range of usual values which are unusual that we would not expect to see.  So “significant” values are what a statistician would call values that are in the tails of our distribution — in other words, those that are not usual, the ones we wouldn't expect to see — whereas the “not significant” ones are the ones that are within that range of usual values.

The criteria that we’re given is that a value x is significantly high or low if the probability of being in that unusual range is less than 1%.  That means we have 1% in each of the tails of our distribution.  That area of 1% in each of the tails of our distribution is a total of 2%, which means 98% of the area under the distribution curve must be within that range of usual values that we would expect, or values that a statistician would call “not significant.”

So if I put down here in my probability field 0.98, and I select the Between option up top, I compute an outcome: the random variables values that I'm looking for.  I'm asked to round to one decimal place.  Fantastic!

Notice again we have 1% in the tail on the left.  This is for values that are significantly low.  And the probability of values being significantly high over here on the right is also 1%.  So 98% is here in the middle, and that's what I've calculated with the Between option from my Normal calculator.

Part 2

Now the second part asks, “Using these criteria, is a back-to-knee length of 26.5 inches significantly high?”  The way you judge this is by looking at whether or not the value you're asked to evaluate is inside the range of usual values, or what a statistician would call values that are “not significant.”  26.5 is within the range of usual values in this problem; it's between 21.3 inches and 26.9 inches.  Therefore, it's in that range of usual values, or what a statistician would call “not significant.”

So here we can say a back-to-knee length of 26.5 inches is not significantly high because it is inside the range of values that are not considered significant.  Good job!

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