18.443, February 22, 2008,  Air pressure as a function of 
       elevation, from the "Engineering toolbox" data:
    
       The elevations in "feet" and "meter[s]" columns are the same 
      except for units. The "feet" are evidently design points, being 
      round numbers, not random variables. Actual measured air pressures 
      are random variables, depending on weather as well as elevation, 
      but in the air pressures as given it seems that weather effects 
      have been averaged out after multiple measurements, and the pressure
      in mmHg (mm. mercury) is standardized as 760.00 at sea level.
      Let's see if we can find how air pressure depends on elevation via 
      regression. First, let's look at simple linear regression. Regressions
      can be done by  R (you aren't required to use R in PS3, however).
      
       Here's the output, where "lm" indicates we're doing a linear 
      regression, of "mmHg" (column header for air pressure in mmHg 
      units) on "feet" of elevation, and "airtable" is the name I gave 
      the table, also in R. For simple linear regression, "Multiple R-squared"
      is just the square of the ordinary correlation coefficient, so r = 
      -0.9284 (it's negative because air pressure decreases with elevation).  
      The most negative residual is < -185 and the largest is > 290 --- too 
      big. In the "Coefficients" table, the last three columns are based on 
      assuming the normal model. The large t and F statistics show the 
      coefficients are both very significantly different from 0, assuming 
      the normal model for errors.  On the next page we'll look at the 
      residuals from the model for a pattern. Here's the R output:
      ------

      Call:
      lm(formula = mmHg ~ feet, data = airtable)

      Residuals:
           Min       1Q     Median     3Q      Max 
        -185.169  -78.586    6.451   81.254  290.234 

      Coefficients:
                   Estimate  Std. Error  t value  Pr(>|t|)    
      (Intercept)  7.083e+02  2.182e+01   32.46   <2e-16 ***
      feet        -9.902e-03  6.431e-04  -15.40   <2e-16 ***
      ---
      Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

      Residual standard error: 114.7 on 38 degrees of freedom
      Multiple R-Squared: 0.8619,	Adjusted R-squared: 0.8582 
      F-statistic: 237.1 on 1 and 38 DF,  p-value: < 2.2e-16