In 1861, Carl Wunderlich averaged more than 1 million measurements of body temperature to conclude that "normal" healthy body temperature was 98.7 degrees Farenheit.  (Of course, the actual measurements were in degrees Celsius.)  However, a recent study published in JAMA (Journal of the American Medical ASsociation), questioned this. Here's the abstract:

JAMA 1992 Sep 23-30;268(12):1578-80
 
  Mackowiak PA; Wasserman SS; Levine MM
  (92389405 NLM)

  OBJECTIVE:To evaluate critically Carl Wunderlich's axioms on clinical
  thermometry.

  DESIGN:Descriptive analysis of baseline oral temperature data from
  volunteers participating in Shigella vaccine trials conducted at the University
  of Maryland Center for Vaccine Development, Baltimore.

  SETTING:Inpatient clinical research unit.

  PARTICIPANTS:One hundred forty-eight healthy men and women aged
  18 through 40 years. MAIN

  MEASUREMENTS:Oral temperatures were measured one to four times
  daily for 3 consecutive days using an electronic digital thermometer.

  RESULTS:Our findings conflicted with Wunderlich's in that 36.8 degrees
  C (98.2 degrees F) rather than 37.0 degrees C (98.6 degrees F) was the mean
  oral temperature of our subjects; 37.7 degrees C (99.9 degrees F) rather than
  38.0 degrees C (100.4 degrees F) was the upper limit of the normal
  temperature range; maximum temperatures, like mean temperatures, varied
  with time of day; and men and women exhibited comparable thermal
  variability. Our data corroborated Wunderlich's in that mean temperature
  varied diurnally, with a 6 AM nadir, a 4 to 6 PM zenith, and a mean amplitude
  of variability of 0.5 degrees C (0.9 degrees F); women had slightly higher
  normal temperatures than men; and there was a trend toward higher
  temperatures among black than among white subjects.

  CONCLUSIONS:Thirty-seven degrees centigrade (98.6 degrees F) should
  be abandoned as a concept relevant to clinical thermometry; 37.2 degrees C
  (98.9 degrees F) in the early morning and 37.7 degrees C (99.9 degrees F)
  overall should be regarded as the upper limit of the normal oral temperature
  range in healthy adults aged 40 years or younger, and several of
  Wunderlich's other cherished dictums should be revised.

  Keywords: Body heat; Temperature;

  MEDLINE RECORD, MDX Health Digest, by Medical Data Exchange

You can view this at http://www.thriveonline.com/health/Library/CAD/abstract6171.html

Unfortunately, I don't have access to either Wunderlich's or Mackowiak et al.'s data, but none-the-less, we're going to do a hypothetical study based on this data.

The Experiment

Suppose we wanted to test for ourselves whether Wunderlich's results were correct.  We might collect 148 subjects and measure their temperature at the same time of day.  If Wunderlich were correct, this would be just like taking  a sample from a larger population of body temperatures whose mean temperature was 98.7 degrees Farenheit..  Let's make some assumptions:

1) Body temperature is normally distributed.
2) The mean body temperature is 98.7 degrees Farenheit.
3) The SD of healthy body temperatures is .8 degrees Farenheit.

Now imagine that I collected body temperatures of 148 subjects.  You can find my measurements here.  (I didn't really do this; I simulated them on a computer much like you did in your last assignment.)   Load this data into an Excel spreadsheet so that we can analyze it.

Goal
Your goal is to determine, based on this data, whether or not Wunderlich was correct.   Some things to keep in mind:
a) I made these data up.  The  data do not necessarily coincide with reality in any way.  Which is to say, maybe I made it up to support Wunderlich, maybe I didn't.
b) Remember from your last lab that even when you know what distribution your sample comes from, it doesn't always look just like that distribution.  In that lab, if you'll remember, you took a sample of 25 observations from a standard normal population, but even though the mean of the population was 0, the average of your sample wasn't necessarily 0.

Theory
Answer these questions before looking at the data.
Let X be a random variable representing the body temperature of a randomly selected human.

1) Is X continuous or discrete?
2) If Wunderlich is correct,  between what two temperatures should we expect about 68% of our sample to lie?
3) What do you expect the average of your sample to be, give-or-take,  assuming Wunderlich is correct?  (Answer this before looking at the data!)

Descriptive
Look at the data.  Describe it using any techniques you think appropriate.  You should imagine yourself speaking to a physician who didn't know that Wunderlich's results were in dispute, and you are trying to describe to her our data.

Inferential
You've probably noticed that the average of our sample isn't equal to 98.7.   There are two competing explanations for this:
a) The mean of the population isn't 98.7
b) Whenever you take a random sample from a population, the average of the sample is always at least a little different from the mean.

Our goal is to decide which of these is the better explanation.  Our method is this;  (b) is the conservative choice, so let's investigate this and see if it seems plausible.  If not, then we'll choose (a).

How can we see if (b) seems plausible?  Here's some of the thinking behind (b):

If I take a random sample from a population with mean m, then the average of the sample should be close to, but not necessarily equal to, m.  So if  the average of our sample is close to 98.7 degrees, then (b) is probably the best explanation for the discrepency we observed between our average and the claimed "normal" body temperature.  However, if our average was FAR from 98.7, then it's probably because it came from a population whose mean is NOT 98.7.

So now the only problem is to determine whether our average is close to or far from 98.7.

How can we determine this?  Later on, we'll have some theory to guide us.  But for now, we can get by with a little experience:

Simulation
1)  Use Excel to generate a random sample of 148 observations from a normal distribution with mean 98.7 and SD .8.  What's the average of this sample?
2) Repeat step (1) a few times (at least 10).  You want to get a feel for the different averages that result.  There's no need to save the actual numbers you generate; just save their averages.

Each time you generated a list of 148 observations, it's as if you actually took a random sample of body temperatures from a group of hypothetical people whose mean body temperature is 98.7.  You now have some idea of how the average of such a sample will vary.  Because we know for a fact that these averages came from a population with mean 98.7, we know that all of these averages are "close to" 98.7.

Questions
1) How many averages did you generate in the simulation?
2) What was the average of your averages?
3) What was the standard deviation of your averages?
4) How many of your averages were above the average of our body temperature data?
5) How many of your averages were below?
6) In your opinion, is the average of our data  "close to" 98.7?  Why?
7) What's your conclusion about "normal" human body temperature?