Question $1(18$ points$)$ Short Answer Questions:

$1)(2$ points$)$ As the sample size increases, the half$-$width of a confidence interval $($A$)$ converges to zero $($B$)$ does not change $($C$)$ converges to infinity $($D$)$ depends on significance level $($ E $)$ first decreases then increase again

$2)(2$ points$)$ The difference between verification and validation is like the difference between: $($A$)$ single and multiple replications; $($B$)$ debugging a program and testing it on a problem with a known solution; $($C$)$ writing new software and updating old software; $($D$)$ being good and being perfect; $($E$)$ they are the same thing

$3)(2$ points$)$ If the probability that a customer's wait in the queue is less than or equal to $10$ minutes is $0\mathrm{.}85,$ then the $\backslash (85^\{\backslash$text $\{$th $\}\}\backslash )$ percentile $($or $0\mathrm{.}85$ quantile$)$ of wait in queue is $($A$)\backslash (1/10\backslash )$ minute $($B$)\backslash (0\mathrm{.}85^\{\backslash$star$\}10=8\mathrm{.}5\backslash )$ minutes; $($C$)10$ minutes; $($D$)$ not obtainable from the information given in this problem; $($ E $)\backslash (1/8\mathrm{.}5\backslash )$ minutes

$4)(2$ points$)$ Suppose that we use common random numbers in order to compare the performance of two systems. If common random numbers is effective, then it should $($A$)$ make the confidence interval shorter; $($B$)$ increase the degrees of freedom; $($C$)$ increase the confidence level; $($D$)$ make the simulation run more quickly; $($E$)$ make the simulation run more slowly

$5)(2$ points$)$ The primary reason we avoid using only a single replication for a steady$-$state simulation is $($A$)$ it is difficult to obtain a confidence interval from within a single replication; $($B$)$ the initial$-$condition bias is worst for a single replication;$($C$)$ we waste too much data with a single replication; $($D$)$ single replication is not applicable for steady state;$($E$)$ all of the above.

$6)(2$ points$)$ In a steady$-$state queueing simulation, if I start the system empty and idle then $($A$)$ I may be fooled by determining the warm up period from one replication; $($B$)$ the system will "warm up$"$ after $100$ customers; $($C$)$ all performance measures will "warm up$"$ in the same amount of time; $($E$)$ determining the warm$-$up period can be objectively determined $($D$)$ all of the above.

$7)(1$ point$)$ True or False We use across$-$replication data to overcome the correlation of consecutive customers.

$8)(1$ point$)$ True or False In a validation study, Increasing the number of replications will decrease the power of the test.

$9)(1$ point$)$ True or False As the sample size increases, a sample mean converges to its true value.

There is a big car wash center with $15$ workers. It operates from $10$am to $8$pm$.$ Since the space is limited$,$ they have a waiting area only up to $5$ cars. If there are already $5$ cars waiting, then new customers $($in our case, cars$)$ will be rejected. System is simulated $10$ times with replication length equal to one day operation. Identify whether each of the following data are within$-$replication data or across$-$replication data:

$10)(1$ point$)($Within or Across$)$ The total number of cars rejected per day

$11)(1$ point$)($Within or Across$)$ The total number of busy workers at time $\backslash ($ t $\backslash )$

$12)(1$ point$)($Within or Across$)$ Wait time of each car in line