Performing hypothesis testing on two independent sample means of soda can fill volumes

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 two independent sample means of soda can fill volumes.  Here's our problem statement: Data on the weights in pounds of the contents of cans of diet soda versus the contents of cans of the regular version of the soda is summarized to the right.  Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal.  Complete Parts A and B below.  Use a 1% significance level for both parts.

Part A1

OK, the first part for Part A says, "Test the claim that the contents of cans of diet soda have weights with the mean that is less than the mean for the regular soda.  The null and alternative hypotheses we can get very easily if we understand that the null hypothesis is by definition a statement of equality.  So therefore we're not going to select Answer option A because here the null hypothesis is not a statement of equality.  The alternative hypothesis will help us to select between the answer options that remain.  And we get the alternative hypothesis typically by reflecting the claim.

In this case, the claim here is that the diet soda weight has an average that is less than the average from the regular soda.  So here in our summary table, we can see that the diet soda is going to be the first sample and the regular soda is going to be the second sample.  So diet is less than the regular, so 1 is going to be less than 2.  So I go down here and look for the answer option that remains with the alternative hypothesis, that 1 is less than 2.  Good job!

Part A2

Now I'm asked for the test statistic.  And notice here our test statistic is a t-score because it says t right here.  So I'm going to open up StatCrunch, and I'm going to resize this window so we can see everything better of what's going on.

OK, so in StatCrunch, now I can go to Stat --> T Stats (because we're looking for that t-score) --> Two Sample (because I've got two samples I'm comparing) --> With Summary (because I'm given a summary table and not actual data).  Here we put in the values for each of the different samples.  So notice here we've got n (which is our sample size), x-bar (which is the sample mean), and then s (which is the sample standard deviation).  So I'm just going to put those values in here for the first sample.  And I'm going to do the same thing for the second sample.

And now this box here for Pooled variances is left unchecked.  You want to leave that alone; it needs to be unchecked.  Down here for our hypothesis test, we need to make sure that this matches what we selected over here.  Notice the difference in the way StatCrunch writes it versus the way it's written in your assignment.  That's OK.  We'll just recognize that they're algebraic equivalents.  Here it's going to be equal to each other.  So when you take a number and subtract the same number from itself, that's going to give you zero.  So we got to leave that claim value alone.  You need to make sure that this inequality sign for the alternative hypothesis matches.

And now we've got everything we need.  And here's our test statistic right here, second to last value in the table in the results window.  So I'm going to put that in here.  I'm asked around to two decimal places.  Nice work!

Part A3

Now, I'm asked for the P-value.  The P-value is located here in the results table, last value.  Notice it says <0.0001; the number is so small --- it's not zero, but it's so small that it might as well be zero.  So that's just what I'm going to put here in my answer field.  Well done!

Now I'm asked to state a conclusion for the test.  The P-value was zero, and with a P-value of zero, you're always going to be inside that critical region, the region of rejection.  Therefore we reject the null hypothesis.  So we're going to choose Answer options B or C.  And every time you reject the null hypothesis, there is always sufficient evidence.  So Answer option C is the one we want.  Fantastic!

Part B1

Now Part B asks for a confidence interval appropriate for the hypothesis test that we just conducted.  I could go through the menu options again, but I'm a little lazy. So I'm going to go back to click Options here on my results window, click on Edit, and it takes me right back to the options window where I flip the radio button over to confidence interval.

I need to put the right level in.  And here it says use a 1% significance level for both parts.  So normally we would say 1% significance level means 99% confidence interval because I'm just subtracting that from 100%.  But we've got two samples.  Therefore, I have to subtract 2α.  So what I really want is 98%.  And here in my results window, you can see the lower and upper limits that I need to put into my answer fields.  So I'm going to do that here.  I'm asked to round to three decimal places.  Well done!

Part B2

And now this final part asks, "Does the confidence interval support the conclusion found with the hypothesis test?"  Well, what does the confidence interval say?  Where is zero with respect to the confidence interval?  Well, zero is not inside the confidence interval.  Zero is outside the confidence interval.  We look at these numbers as though they're on a number line.  Zero is going to be here to the right outside the confidence interval.  Therefore, there's going to be a difference between the two main values.  They're not the same, but that's what the null hypothesis here says.  The null hypothesis says they are the same.  We have evidence that they're not the same.  Therefore we're going to reject this statement because it's false.

That's the same thing that we got here from the hypothesis test --- rejected the null hypothesis.  So we're going to say, yes, the confidence interval does support the conclusion from the hypothesis test.  They’re the same conclusion, because the confidence interval contains in this case only negative values.  And you can see both our upper and lower limits are negative.  So only negative numbers are going to be in between.  Fantastic!

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