Question

Statistics Out- of- Control Signals

Out-of-control signal I: Any point falls beyond the ±3σ level.

Out-of-control signal II: A run of nine consecutive points on the same side of the center line.

Out-of-control signal III: At least two of three consecutive points lie beyond the ±2σ level on the same side of the center line.

Yellowstone Park Medical Services provides emergency health care for park visitors. Such health care includes treatment for everything from indigestion and sunburn to more serious injuries. A recent issue of Yellowstone Today indicated that the average number of visitors treated each day was 21.7. The estimated standard deviation was 4.2.

For a ten- day summer period, the following data were obtained:

Day 1 2 3 4 5 6 7 8 9 10

Number Treated 20 15 12 21 24 28 32 36 35 37

The manager of a motel has 316 rooms. From observation over a long period of time, she knows that on an average night, 268 rooms will be rented. The long-term standard deviation is 12 rooms. For 10 consecutive nights, the following numbers of room were rented each night:

Night 1 2 3 4 5 6 7 8 9 10

Number of Rooms 238 245 261 269 273 250 241 230 215 217

+3σ = 34.3…………………………………………………………………………………………………………

+2σ= 30.1……………………………………………………………………………………………………….

+1σ = 25.9………………………………………………………………………………………………………..

µ = 21.7 _________________________________________________________

-1σ= 17.5………………………………………………………………………………………………………………

-2σ= 13.3………………………………………………………………………………………………………………

-3σ =9.1 ………………………………………………………………………………………………………………

Days 1 2 3 4 5 6 7 8 9 10

+3σ = …………………………………………………………………………………………………………

+2σ= ……………………………………………………………………………………………………….

+1σ ………………………………………………………………………………………………………..

µ = _________________________________________________________

-1σ= ………………………………………………………………………………………………………………

-2σ= ………………………………………………………………………………………………………………

-3σ = ………………………………………………………………………………………………………………

Nights 1 2 3 4 5 6 7 8 9 10

Answer 1

R code:

x=c(20,15,12,21,24,28,32,36,35,37)
n=length(x)

a=c(1:n)
mu=21.7
sig=4.2
UCL=replicate(n,mu+3*sig)
CL=replicate(n,mu)
LCL=replicate(n,mu-3*sig)

plot(a,x,type="b",ylim=c(5,42),xlim=c(-1,12),xlab="Observation")
lines(a,UCL,type="l",text(0,34.3,expression(mu+3*sigma)),text(11,34.3,"UCL"))
lines(a,replicate(n,mu+2*sig),type="l",text(0,30.1,expression(mu+2*sigma)))
lines(a,replicate(n,mu+sig),type="l",text(0,25.9,expression(mu+sigma)))
lines(a,CL,type="l",text(0,21.7,expression(mu)),text(11,21.7,"CL"))
lines(a,replicate(n,mu-2*sig),type="l",text(0,13.5,expression(mu-2*sigma)))
lines(a,replicate(n,mu-sig),type="l",text(0,17.5,expression(mu-sigma)))
lines(a,LCL,type="l",text(0,9.1,expression(mu-3*sigma)),text(11,9.1,"LCL"))

Graph:

Hence

Hence there is a out of control signal II and signal I for the last 3 points.

R code:

x=c(238,245,261,269,273,250,241,230,215,217)
n=length(x)

a=c(1:n)
mu=268
sig=12
UCL=replicate(n,mu+3*sig)
CL=replicate(n,mu)
LCL=replicate(n,mu-3*sig)

plot(a,x,type="b",ylim=c(210,310),xlim=c(-1,12),xlab="Observation")
lines(a,UCL,type="l",text(0,UCL,expression(mu+3*sigma)),text(11,UCL,"UCL"))
lines(a,replicate(n,mu+2*sig),type="l",text(0,mu+2*sig,expression(mu+2*sigma)))
lines(a,replicate(n,mu+sig),type="l",text(0,mu+sig,expression(mu+sigma)))
lines(a,CL,type="l",text(0,CL,expression(mu)),text(11,CL,"CL"))
lines(a,replicate(n,mu-2*sig),type="l",text(0,mu-2*sig,expression(mu-2*sigma)))
lines(a,replicate(n,mu-sig),type="l",text(0,mu-sig,expression(mu-sigma)))
lines(a,LCL,type="l",text(0,LCL,expression(mu-3*sigma)),text(11,LCL,"LCL"))

Graph:

Hence there is a out of control signal II and signal I for the last 3 points.