Applying a nonstandard normal distribution to quality control

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 quality control.  Here's our problem statement: The weights of a certain brand of candies are normally distributed with a mean weight of 0.8596 grams and a standard deviation of 0.0512 grams.  A sample of these candies came from a package containing 445 candies, and the package label stated that the net weight is 380.1 grams.  If every package has 445 candies, the mean weight of the candies must exceed 0.8541 grams for the net contents to weigh at least 380.1 grams.

Part A

Part A says, "If one candy is randomly selected, find the probability that it weighs more than 0.8541 grams."  Well, to do this, I'm going to use the distribution calculators in StatCrunch, because here it says that my weights are normally distributed.  So I want that normal distribution calculator.  Alternatively, you can use the z-score tables to calculate this out, but hey, it's the 21st century.  Let's join it.  Alright, so I'm going to open up StatCrunch.  Not that I need StatCrunch per se, but the distribution calculators that I want to use are actually here inside StatCrunch.  I'll resize this window so we can see better what's going on here.

OK, so to get my calculator, I go to Stat --> Calculators --> Normal, because again it --- the problem statement said that our weights are normally distributed.  So here's my normal calculator in StatCrunch.  The nice thing about this is that I don't have to do any adjustment with the z-scores like I would with the tables.  I just go ahead and put the mean and the standard deviation here into my calculator and StatCrunch handles all that conversion stuff for me.  Again, it's nice to join the 21st century.

So the mean weight that we get from the problem statement is up here, 0.8596, so I'm going to stick that here in the mean field.  The standard deviation from the problem statement is 0.0512.  So I stick that there.  And I want the probability that we have a weight more than 0.8541, so 0.8541, and I want to get more than.  And there's my probability, calculated out for me.  Well done!

Part B

Now Part B asks us the same question, only this time instead of one candy being randomly selected, we're randomly selecting 445 candies.  So everything's pretty much going to remain the same here in StatCrunch, except we need to make an adjustment for our standard deviation.

Remember that when you've got more than one that you're selecting and you're looking for that probability, you have to make an adjustment to your standard deviation because the standard deviation is a biased estimator.  So you've got to make some adjustments to account for that bias.  We're not going to make any adjustments with the mean value because the mean value is an unbiased estimator.  So there's no bias to account for with this.  So we're going to leave the mean value alone.

We have to make an adjustment with the standard deviation, and the adjustment we make is to divide the standard deviation by the square root of whatever our new sample size is, in this case, 445.  So I take my calculator out here, and I'm going to actually perform that operation.  The standard deviation from the problem statement, 0.0512, and I'm going to divide you by the square root of the 445.  There's my new standard deviation.

So I could transfer all these numbers manually and be here, but I'm prone to transcription errors.  So I'm just going to take this, and I'm just going to copy that number from the calculator, come here to StatCrunch, select everything in the field, and Ctrl+V on my keyboard will paste that bad boy number right in there for me.  So I don't have to worry about transcription errors.  And now I just press Compute! again, and voila!  There's the new probability.  Nice work!

Part C

Now Part C asks, "Given these results, does it seem that the candy company is providing consumers with the amount claimed on the label?"  Well, the result you really want to focus in is the one we just got done calculating because each package contains the 445 candies that we just calculated the probability for.

And look, that probability is really close to 1.  So it's almost certain that we're going to be providing what's on the label.  So then are, you know --- is the candy company providing consumers with the amount claimed on the label?  The answer's going to be yes, because your probability is just about 1.  And so, yes, because the probability of getting a sample mean when 445 candies are selected is not exceptionally small because, look, it's almost 1, which means it's exceptionally large.  I check my answer.  Excellent!

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