Interval Estimate for a Fixed Number of Replications

• The sample mean can be calculated as above
• The sample variance
  
  $$\hat{\sigma}^2 (\hat{\theta}) = \frac{1}{R} ~ \sum_{r=1}^R \frac{(\hat{\theta_r} - \hat{\theta})^2} {R-1}$$
  
  

• When the output data are of the form $\{ Y_r(t), 0 \le t \le T_E \}$ for r = 1, 2, ..., R

  
  

  $$\hat{\phi_r} = \frac{1}{T_E} \int_{0}^{T_E} Y_r (t) dt, ~~ r = 1,2,...,R$$
  
  
  
  
  $$\hat{\phi} = \frac{1}{R} \sum_{r=1}^R \hat{\phi_r}$$
  
  
  and
  
  $$\hat{\sigma}^2 (\hat{\phi}) = \frac{1}{R} ~ \sum_{r=1}^R \frac{(\hat{\phi_r} - \hat{\phi})^2} {R-1}$$
  
  

• The sample variance is essentially a measure of error. In statistics, we use its square root as the standard error.

• Example 12.10 on page 446 (The Able-Baker carhop problem, continued): calculate the confidence interval for Able's utilization and average waiting time.

• Example 12.11 on page 447: the more runs, the more accurate the result is.