Finding probabilities for different sample sizes using a nonstandard 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 probabilities for different sample sizes using a nonstandard normal distribution.  Here's our problem statement: The overhead reach distances of adult females are normally distributed with a mean of 200 centimeters (cm) and a standard deviation of 8 cm.

Part A

Part A says, “Find the probability that an individual distance is greater than 212.5 cm.”  This problem is very easily tackled inside StatCrunch, so I’m gonna pull up StatCrunch here.  And inside StatCrunch, I'm going to go to Stat –> Calculators –> Normal.  Why am I pulling up the Normal calculator?  Well, I'm doing that because the problem statement says that our data are normally distributed.

Note that in the Normal calculator, the mean and standard deviation values by default come up for the standard normal distribution.  The problem statement gives us a nonstandard normal distribution.  But this is very easily corrected.  I just adjust the mean and the standard deviation for the values that are listed in the problem statement.

Then in the line below we see a probability calculation.  We know this is so because the P on the left here signals probability.  So this is the probability that x is greater than or less than a given random variable value.  That's going to be equal to a probability value or an area under the curve.  Here the problem statement asks us to find the probability for an individual distance greater than 212.5 cm.  So down here, I'm going to put “greater than” from the drop down menu, 212.5 for the random variable, and click on Compute!  StatCrunch automatically computes my probability for me.  I'm asked around to four decimal places, so i do that.  Good job!

Part B

Now Part B asks, “Find the probability that the mean for 20 randomly selected distances is greater than 198.2 cm.”  Now before you rush off and just change this random variable value here in your Normal calculator in StatCrunch, you need to make an adjustment to your standard deviation.  Why?  Because you're taking a sample that's greater than one in size.

The adjustment that we need to make is shown here.  We take the population standard deviation (that's given to us in the problem) and we divide it by the square root of the sample size.  This is an adjustment that we need to make to the standard deviation.  So I just plug in the values that's given to me in the problem statement.  I have a standard deviation of 8, and the sample size of 20, so 8 divided by the square root of 20 gives me 1.7889 (rounded to 4 decimal places).

I could just put this new value for standard deviation into my calculator in StatCrunch.  However, because Pearson is often so exacting with the precision of answers that they want you to provide in the answer fields for your assignments, I'm a little weary of putting any sort of rounded number into the calculator and then getting another number that I have to round out.  It might be off just enough to where Pearson marks me wrong.

So what I'm going to do is make this same calculation with a calculator that's inside my computer.  Then I can copy the value from the computer and paste it into the calculator in StatCrunch.  So take 8 divided by the square root of 20.  That gives me the same value, but look at all these decimal places that I get to put into my value.  I right-click on the mouse, select Copy, and then back into StatCrunch, come here to Standard Deviation, and press Ctrl+V on my keyboard, and this puts that entire number into the field in StatCrunch.  I press the Home button on my keyboard so I can go to the front of that field so I can see that the same number is actually in there.

Now all I need to do is make the appropriate adjustment to my random variable value.  Here in the problem statement, we want the probability that the mean of 20 randomly selected instances is greater than 198.2, so I make that adjustment here.  Press Compute! and out comes my probability.  I want to round to four decimal places, so I do that, put my answer in, press Enter or Check Answer.  Excellent!

Part C

And now the last part, Part C, asks, “Why can the normal distribution be used in Part B even though the sample size does not exceed 30?”  Well, notice that this last part of the question — “even though the sample size does not exceed 30" --- is a reference to the Central Limit Theorem, which says that if our sample size is greater than 30, we can assume that our data conforms to a normal distribution.

However, the Central Limit Theorem also says that there's no threshold value to me for sample size if we're already normally distributed.  That makes sense, right?  I mean, why would you assume that you're normally distributed?  We already have a normal distribution; there's no need for that extra assumption.  You're already normally distributed.  And that's the answer to the question here in Part C.  Why can the normal distribution be used?  Because we're already normally distributed.  It says so right here in our problem statement — we're normally distributed.  Therefore that's the answer option that I want to select.  And looking at the different answer options available to me, I can see that Answer Option B is the one that I want to select.  Well done!

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