Finding and interpret a confidence interval for a population mean (σ unknown)

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 and interpret a confidence interval for a population mean when the population standard deviation is unknown.  Here's our problem statement: A clinical trial was conducted to test the effectiveness of a drug for treating insomnia.   In older subjects before treatment, 24 subjects had a mean wake time of 102 minutes.  After treatment, the 24 subjects had a mean wait time of 96.4 minutes and a standard deviation of 43.7 minutes.  Assume that the 24 sample values appear to be from a normally distributed population, and construct a 90% confidence interval estimate of the mean wake time for a population with drug treatments.  What does the result suggest about the mean wake time of 102 minutes before the treatment?  Does the drug appear to be effective?

Part 1

OK, the first part of this problem asks us to “construct the 90% confidence interval estimate of the mean wake time for a population with the treatment.”  This is most easily done inside StatCrunch, so I'm gonna pull up StatCrunch.

Notice I have StatCrunch already set up in a separate window.  I could come over here in Question Help and then select StatCrunch from the list, but then the window that pops up for StatCrunch would be confined inside this window where I need to put my answers.  So in order to clear the field so I can see my work and then put answers in the answer field, I like to have StatCrunch set up in a separate window.

Here in this separate window, we don't actually need any actual data.  We have summary statistics listed here in the problem statement, and so we're going to use those to construct our confidence interval.

However, before we get into StatCrunch, because we're asked to find a confidence interval estimate for the mean, we need to ask ourselves the key question: Do we know what the population standard deviation is?  The answer to that question is no.  We have listed here in the problem statement a standard deviation value, but that's for the sample of 24 subjects.  We don't have a standard deviation value given for the population.  Therefore, we don't know what it is.  And therefore we want to use the Student-t distribution to calculate our confidence interval estimate.

If we knew what the population standard deviation was, then we would use the standard normal distribution and calculate z-scores to get our confidence interval estimate.  But here we don't know what it is (and that's typically the case), so we're going to use the Student-t distribution.  Therefore, up here in StatCrunch, I want to select Stat –> T Stats (because we're using the Student-t distribution) –> One Sample (because we only have one sample for our summary statistics that we're looking at), and then I don't have any actual data to put into StatCrunch, so I'm gonna select With Summary.

Here in the options window, I need to put in the summary statistics that I found in the problem statement.  So the mean wake time that we want is for the group of the treatment because that's what we're asked to make the confidence interval estimate for — a population with the treatment.  So here after treatment, the 24 subjects had a mean wake time of 96.4 minutes. So here in this first field, I'm gonna put 96.4, and then in the next field the standard deviation value, and then the sample size in this last field.  Then I'm going to come down here and select the radio button for Confidence interval and make sure the confidence interval levels match what I'm looking for.

Then I just press Compute!, and out pops this lovely results window with the upper and lower limits for my confidence interval.  So I just put those here into my answer field.  I check my answer.  Nice work!

Part 2

Now, the second part of the problem asks me, “What does the result suggest about the mean wake time of 102 minutes before the treatment?  Does the drug appear to be effective?”  To evaluate this, we need to take this value we're asked to evaluate — 102 minutes — and see where it lies with respect to our confidence interval estimate.

Is 102 inside the confidence interval or outside the confidence interval?  Here 102 is inside our confidence interval.  Therefore, it could potentially be the mean value of those who actually have the treatment for the population.

Remember this mean time that we have up here — 96.4 — is for our 24 subjects in the sample.  We want to know about the population.  So the sample mean is 96.4, but the population mean could be anywhere between 81.1 and 111.7.  102 is inside our confidence interval.  Therefore, it could potentially be the mean, and therefore if it's the same mean, then there's no change from what we had previously before applying the drug treatment.  Therefore, the drug doesn't seem to pose any significant effect.  If, on the other hand, the value we’re asked to evaluate is outside the confidence limit, well, then it seems to have some sort of significant effect.

But that's not the case here.  So the confidence limit interval here actually includes the mean wake time, so that means before and after the treatment could potentially be the same.  This means that the drug treatment does not have a significant effect.  Check my answer.  Good job!

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