Confidence Intervals

1
Confidence Intervals

• Underlying model Unknown parameter
• We know how to calculate point estimates
• E.g. regression analysis
• But different data would change our estimates.
• So, we treat our estimates as random variables
• Want a measure of how confident we are in our
  estimate.
• Calculate Confidence Interval

2
What is it?

• If know how data sampled
• We can construct a Confidence Interval for an
  unknown parameter, q.
• A 95 C.I. gives a range such that true q is in
  interval 95 of the time.
• A 100(1-a) C.I. captures true q
• (1-a) of the time.
• Smaller a, more sure true q falls in interval,
  but wider interval.

3
Example 1 Lead in Water

• Lead in drinking water causes serious health
  problems.
• To test contamination, require a control site.
• Problems
• Lead concentration in control site?
• Estimate 95 confidence interval

4
Example 2 Gas Market

• Recall U.S. gas market question
• By how much does gas consumption decrease when
  price increases?
• Our linear model
• Estimate of b1 -.04237.
• How confident are we in this estimate?
• Construct 90 C.I. for this estimate

5
If Data N(m,s2)

• Since we dont know s, use t-distribution.
• 95 C.I. for m
• s is standard error of mean.
• t97.5 is critical value of t distribution
• Draw on board (Prob 2.5)

6
t-distribution

• Similar to Normal Distribution
• Requires degrees of freedom.
• df ( data points) ( variables).
• E.g. mean of lead concentration, 8 samples, one
  variable d.f.7.
• Higher d.f., closer t is to Normal distribution.

7
If Distribution Unknown

• Can use Bootstrapping.
• Draw large sample with replacement
• Calculate mean
• Repeat many times
• Draw histogram of sample means
• Calculate empirical 95 C.I.
• Requires no previous knowledge of underlying
  process

8
Lead Concentration

• 8 lead measurements
• Mean51.39, s5.75, t97.52.365
• Lower51.39-(5.75)(2.365)
• Upper 51.39(5.75)(2.365)
• C.I. 37.8,65.0
• Using bootstrapped samples
• C.I. 40.8,62.08

9
Gas Regression S-Plus

• Coefficients
• Value Std. Error t value
  Pr(gtt)
• (Intercept) -0.0898134 0.0507787 -1.7687217
  0.0867802
• PG -0.0423712 0.0098406 -4.3057672
  0.0001551
• Y 0.0001587 0.0000068 23.4188561
  0.0000000
• PNC -0.1013809 0.0617077 -1.6429209
  0.1105058
• PUC -0.0432496 0.0241442 -1.7913093
  0.0830122
• Residual standard error 0.02680668 on 31 degrees
  of freedom
• Multiple R-Squared 0.9678838
• F-statistic 233.5615 on 4 and 31 degrees of
  freedom, the p-value is 0

10
Gas Price Response

• b2-.04237, s.00984
• 90 C.I. t951.695 (d.f.37-532)
• C.I. -.0591,-.0256
• Using bootstrapped samples
• C.I. -.063,-.026
• Response is probably between 2.5 gallons and 6
  gallons.

11
Interpretation Other Facts

• There is a 95 chance that the true average lead
  concentration lies in this range.
• There is a 90 chance that the true value of b1
  lies in this range.
• Also can calculate confidence region for 2 or
  more variables.