My question is, can we use control chart to evaluate the process, because the laboratory test should be done within a specific timeframe?
The answer is yes, but....
Theorically you are right, but your data looks a little bit strange, it looks like the difference of the time the doctor order laboratory test and the time the lab technician arise the result from system is time dependient
(or day to say so), the time has some efect that in the first days looks like they do the complicated job, and then after finished with the complicated ones, they do the normal job ( the last part of the chart).
Difference in time = -5.46667 * (time in days since the beginning) + 106.6
rsq = 0.882125
Weir, IMHO, I would like to know why such behabiur, remember that the control charts work for the same process condition, you can't mix different things in one chart.
Of course you can use the slope to estimate values and take off the trend, so it could be more meaninfull the control chart (but the differences look to much autocorrelated as intervals for 4 points cicles).