Conducting mean hypothesis testing on back pain treatment data

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to conduct mean hypothesis testing on back pain treatment data.  Here's our problem statement: Researchers conducted a study to determine whether magnets are effective in treating back pain.  The results are shown in the table for the treatment with magnets group and the sham or placebo group.  The results are a measure of reduction in back pain.  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 0.05 significance level for both parts.

Part A1

OK, we can see here that the data that we need to run our hypothesis test is here in this table.  And notice that the treatment group is Group 1 and the placebo group is Group 2.  This corresponds with the usual practice that you see here in the problem statement where the one that's mentioned first is Group 1 and the one that's mentioned second is Group 2.  So here we have the first part to Part A, which says, "Test the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment.  What are the null and alternative hypotheses?"  Well, here you can see that the treatment group was Group 1.  The sham or placebo group is Group 2.  So we're testing the claim that Group 1 has a greater mean than Group 2.

What's the null alternative hypothesis that we want?  Well, the null hypothesis is always a statement of equality.  So looking at my answer options here, I don't want Answer options B or D because those null hypotheses are not statements of equality.  How do we choose between Answer options A and C?  Well, we look at the alternative hypothesis.  Remember we're testing the claim that Group 1 has a greater mean than Group 2.  And look at my alternative hypothesis here.  That's reflected here in Answer option C.  Well done!

Part A2

Now the second part of Part A asks for the test statistic.  Notice it's a t score.  So when I go into StatCrunch to get my hypothesis testing, I know I need to look for a t score.  Let's resize this windows so we can see better what's going on here.  And now to get my hypothesis test, we go to Stat --> T Stats (because we want a t score) --> Two Sample (because we have two samples we're comparing) --> With Summary (because we don't have any actual data).

Here in my options window, it asks me for the summary stats.  And these are the same numbers that come out of the table that we see here in the problem statement.  So I'm going to go ahead and type in those numbers.  The sample mean is x-bar, so here we have the sample mean for the first group.  Sample standard deviation is s.  And the sample size is n.  Let me go ahead and enter in those same numbers for the second sample.  And then down here, notice the default selection on the radio button here is for hypothesis testing, so we'll leave that alone.  Notice that the area here needs to match what we have here, but it's organized differently in StatCrunch than it is in your assignment.  So if we know here, from what we've actually selected as the right answer, that our null hypothesis says that mu1 equals mu2.  Here we've got mu1 minus mu2.  Well, if mu1 and mu2 are the same number and we subtract them, we're going to get zero.  So we're going to leave this claimed value alone.  Then here in this field, we need to make sure our inequality sign matches what we have here.  And now we've got everything we need.  So we hit Compute!, and our test statistic [is the] second to last value there in the results window table.  I'm asked to round to two decimal places.  Fantastic!

Part A3

Now the next part asks for the P value.  That's the last value listed here in the results window table, right next door to the test statistic.  I'm asked to round to three decimal places.  Excellent!

Part A4

Now the next part for Part A says, "State the conclusion for the test."  To do this, let's compare the P value with our significance level.  Here our P value is almost 42% [with a] significance level of 5%.  So our P value is way larger than our significance level, which means we're outside the region of rejection.  And whenever you're outside the region of rejection, you're going to fail to reject.  And every time you fail to reject, there is not sufficient evidence.  Good job!

Part A5

And now the last part of Part A asks, "Is it valid to argue that magnets might appear to be effective if the sample sizes are larger?"  Well, if we have a larger sample size, it's going to be easier for the test to detect a statistically significant difference.  Remember, there's a difference between practical significance and statistical significance.  And what we're testing with the hypothesis test is statistical significance.  With a larger sample size, you might actually get, you know, more --- I guess a more sensitivity to detecting that statistical significant difference.  Let's look at our drop downs here to see what we could select from that.

Here we're at the first one.  We're asked to select between the mean and the standard deviation.  We're going to choose the mean because that's what our hypothesis test is testing.  It's testing a difference in the population means.  So we want to be looking at the sample means for comparison.  The second drop down asks us to compare the sample means for each of the two different groups.  So we look at here, x-bars, our sample mean, and we see that the treatment group has a greater mean than the placebo group.  And then finally, we just got done saying that it is valid to argue that magnets might appear to be effective, because with the larger sample size you're more likely to detect the statistical significant difference and that's what you need in order for the test to come out to say that, yeah, the magnets are going to be effective.  Good job!

Part B

And now Part B of this problem asks us to construct a confidence interval.  Well, that's very easily done.  If we go back here to our results window in StatCrunch, I can click on Options --> Edit.  That takes me back to my options window.  I scroll down here, switch that radio button from hypothesis test to confidence interval, I need to make sure I put in the right level here --- we've got a 5% significance level, but remember we've got two samples now with a one tailed test, so that means I need to take out two alpha, not just one.  So that 95% now becomes 90%.  And here are my lower and upper limits for my confidence interval.  I'm asked to round to three decimal places, so I will do that here.  There's my lower limit.  And now here is my upper limit.  Excellent!

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