Performing hypothesis testing on soda can fill variation

Intro

Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to perform hypothesis testing on soda can fill variations.  Here's our problem statement: Workers at a certain soft drink factory collected data on the volumes in ounces of a simple random sample of 23 cans of the soda drink.  Those volumes have a mean of 12.19 ounces and a standard deviation of 0.09 ounces, and they appear to be from a normally distributed population.  If the workers want the filling process to work so that almost all cans have volumes between 11.93 ounces and 12.53 ounces.  The range rule of thumb can be used to estimate the standard deviation should be less than 0.15 ounces.  Use the sample data to test the claim that the population of volumes has a standard deviation less than 0.15 ounces.  Use a 2.5% significance level.  Complete Parts A through D below.

Part A

Alright, Part A says, "Identify the null and alternative hypotheses."  So we know that the null hypothesis is by definition a statement of equality, so we're not going to choose Answer option A or Answer option D.  To select between Answer options B and C, we look at the alternative hypothesis that typically comes from the claim.  And here the claim is that the population of volumes has a standard deviation less than 0.15 ounces.  So our sigma, which is the standard deviation for the population, is going to be less than 0.15 ounces.  We want Answer option B.  Nice work!

Part B

Now Part B says, "Compute the test statistic."  We can get the test statistic by performing our hypothesis test in StatCrunch.  So I'm going to pull up StatCrunch here.  And I'll pop that window out, and then I'm going to resize this window so we can see a little bit better what's going on here.  OK, inside StatCrunch, I want to go to Stat --> Variance Stats (because this is how we test standard deviation; it's through the variance) --> One Sample (because I've got only one sample) --> With Summary (because I don't have actual data).

Here in the options window, I'm first asked for the sample variance.  Well, look here in the problem statement, and we see that we're not given the variance, but we are given the standard deviation.  If I square standard deviation, that gives me the variance.  The variance is the square of the standard deviation.  So if I pull out my calculator here and I take that standard deviation of 0.09 and I square it, that gives me the variance for my sample, which is 0.0081.  My sample size is the 23 cans.  My hypothesis test --- I want to make sure that this matches what we have over here, but notice how what we have here in our problem statement, the answer we selected, is a hypothesis test on standard deviation.  Here in StatCrunch, we're looking at variance.  So in order for these to match up, I've got to take this claimed value and I've got a square it.

So let's do that.  I pull back my calculator, and that was 0.15.  I square it, and that's the number you want to stick in here for your claimed value for your hypothesis test.  I need to make sure that this inequality sign matches here for my alternative hypothesis.  And now I'm ready to go.  I hit Compute!, and here's my chi-square test statistic, which I can put here in my answer field.  Fantastic!

Part C

Now, Part C asks for the P-value, which of course is the last value in that results window table right next door to the test statistic.  Good job!

Part D

And finally, Part D says, "State the conclusion."  Well, here we have a P-value of less than 1%, and we're comparing that with a significance value of 2.5%.  So our P-value is going to be less than our significance level, and that means that we're inside the region of rejection.  When you're inside the region of rejection, you reject the null hypothesis.  And every time we reject the null hypothesis, there's always sufficient evidence.  Good job!

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