This lab analyzes a data sets from Weisberg's book
  <CITE>Applied Linear Regression.</CITE> Weisberg writes:

<BLOCKQUOTE> The data in this example are deduced from a diagram in a
paper written by W. Stanley Jevons (1868), and provided by Stephen
M. Stigler.  In a study of coinage, Jevons weighed 274 gold sovereigns
that he had collected from circulation in Manchester, England.  For
each coin, he recorded the weight after cleaning to the nearest .001
gram, and the date of issue. ... The standard weight of a gold
sovereign was supposed to be 7.9876 grams; the minimum legal weight
was 7.9379 grams.  </BLOCKQUOTE>

The Jevons article gives summary statistics, not the complete data
set.  The S-Plus commands below will generate a data set for you that
approximates Jevons' data.

 <P>
<B>
age _ 1:5 # age in decades<BR>
number _ c(123,78,32,17,24) # sample size in each decade<BR>
ave.decade _ c(7.9725,7.9503,7.9276,7.8962,7.8730) # average weight in each decade<BR>
SD.decade _ c(0.01409,0.02272,0.03426,0.04057,0.05353) # SD in each decade<BR>
x _ rep ( age, number )<BR>
y _ rnorm ( length(x),  rep(ave.decade,number), rep(SD.decade,number) )
</B>
 <P>
Plot the data; add a line showing the minimum legal weight<BR>
<B>
plot ( x, y, xlab="Age in Decades", ylab="Weight in Grams" )<BR>
abline ( h=7.9379 )<BR>
</B>
If you prefer a dashed, dotted or other kind of line, then remake your
plot, but use <B>abline ( h=7.9379, lty=2 )</B>.  "lty" stands for "line type".
Try lty=2, lty=3, etc. to see what kinds of lines you can get.
 <P>
# Find the regression line<BR>
# We need to know averages, SD's and correlation for X and Y<BR>
<B>
ave.x _ mean(x)<BR>
ave.y _ mean(y)<BR>

SD.x _ sqrt ( mean ( (x-ave.x)^2 ) )<BR>
SD.y _ sqrt ( mean ( (y-ave.y)^2 ) )<BR>

r _ cor(x,y)<BR>
</B>
# The line goes through the point (ave.x,ave.y) and has slope r*SD.x/SD.y.<BR>
# Add it to the plot.<BR>
<B>
slope _ r*SD.y/SD.x<BR>
intercept _ ave.y - slope*ave.x<BR>
abline(intercept,slope,lty=2)<BR>
</B>
 <P>
The standard weight of a new coin is 7.9876 grams.  Is the regression
line consistent with the standard weight?

The question asks about coins that are 0 decades old.  What does our
regression line say about the average weight of such coins?
 <P>
The minimum legal weight for gold coins was 7.9379 grams
Two interesting questions:  What percentage of new coins are below
the minimum; what percentage of coins 5 decades old are below
the legal limit?
 <P>

SD.decade gives the SD of weight for each decade separately.  Would
you expect the SD of y to change over time?  Plot SD.decade against
time.  Is this what you thought it would be?  Can you think of an
explanation?  Can you estimate what the SD of new coins would be?