So we know the following from last time:

1

• So we know the following from last time
• and that the standardized estimators (take away
  the mean of the estimator, divide by the standard
  error of the estimator) have t-distrbutions with
  (n-2) degrees of freedom. Thus we may test
  hypotheses about the true regression parameters
  and m0 or we may construct confidence intervals
  for them.
• Go over each of these for the Hardness data...

2

• Section 2.6 deals with another instance of
  estimation, this time the prediction of a single
  observation, yp, resulting from a new independent
  value of the explanatory variable, xp , assuming
  the same regression model holds.
• Note the distinction between estimating the mean
• and
  predicting a new response yp in the former
  case were estimating the mean for all responses
  at xp and in the latter case were only
  estimating the response for a single individual
  outcome. The key here is that the single
  observation is much more variable than the mean
  so when we construct confidence bands around the
  prediction line, the latter bands (called
  prediction bands) will be much wider than the
  former (called confidence bands)...

3

• The standardized yp is given at the top of p.40
  and when using the standard error instead of the
  standard deviation in the denominator, we get a
  t(n-2) distribution as before so we may test
  hypotheses or set up prediction intervals using
  this information. 2.29 on page 40 shows that
  interval. Note that the interval is wider than
  the confidence interval for the mean mp at xxp
  since the s.e. in the former is gt s.e. in the
  latter
• Check these out in R3... can you plot the
  confidence bands and prediction bands?

4

• Assignment for Tuesday, Jan. 31
• Put your chosen dataset into a data frame in R
• Do a scatterplot
• Do least squares regression to find the estimates
  of the slope and intercept parameters. Interpret
  these values in context.
• Plot the prediction line along with both
  confidence bands and prediction bands
• Try to program R to do the computations in the
  formulas weve been working with, instead of
  through its function lm.
• Finish reading the text up through section 6,
  page 41. Well review on Thursday and begin
  section 2.7
• Redo problems 2.17 and 2.18 complete those
  parts that weve now covered...
• Add problems 2.25 2.26 and complete the parts
  weve covered in class...