Practice Problems in X , R and  charts
(Use formulae given in slide nos. 17,18,19 and 20 in Third Lecture).
(Use tables 1,2,3 and 4 for various factors).
Q. 1. Control charts for X and R are maintained on the tensile strength in
pounds of a certain yarn. The subgroup size is 5. The values of X and R
are computed for each subgroup. After 25 subgroups,  X = 514.8 and
R
= 120.0 Compute the values of 3-sigma limits for X and R charts,
and estimate the value of process dispersion on the assumption that the
process is in statistical control.
(σ΄=2.06)
Q. 2. Control charts for X and  are maintained on the weight in ounces
of the contents of a certain container. The subgroup size is 10. The values
of X and  are computed for each subgroup. After 18 subgroups,  X
= 595.8 and
  = 8.24. Compute the values of 3-sigma limits for the
X
and  charts and estimate the value of process dispersion on the
assumption that the process is in statistical control. (σ΄=0.496)
Q. 3. Control charts for X , R and  are to be maintained on the drawings
from a bowl of chips the distribution of which is approximately normal.
The subgroup size is 5. X  is 60, and  ' is 8. Assume that 3-sigma
control limits are to be based on X  and  ', compute the values of the
upper control limit, central line and the lower control limit for the X , R and
 charts respectively.
Q.4. In order to meet government regulations, the contained weight of a
product must equal or exceed the labeled weight 99.9% of the time.
Control charts for X and  are maintained on the weight of the contents
using a subgroup size of 10. After 20 subgroups, X =731.4 and 
9.16. Estimate the value of  assuming that the process is in statistical
control. If the label weight is 35 (lower sp. Limit), and assuming the
process generates a normal distribution, does it meet the federal
requirements? What is AOF? YES (σ΄=0.496, Area below L=0.08%), 1.57
Q.5. X and R charts have been maintained on a certain quality
characteristic. All points have fallen within the control limits on both the
charts. A sudden change in the process occurs that increases X  by
1.5 but does not change  . In answering the following questions,
assume that the quality characteristic is normally distributed both before
and after the change, and that the control limits are based on
observations made before the shift in process centering.
a) If the subgroup size is 3, approximately what percentage of points
would you expect to fall outside control limits on the  chart because of
the change in  ? What percentage on the R chart? Also find ARL.
(Draw distribution of  and find area above UCL after the shift), 34.4%
(No effect on R chart since there is no change in dispersion), ARL=2.906
b) Answer the same questions assuming a subgroup size 5. (64%)
c) Answer the same questions assuming a subgroup size 8. (89%)
Q. 6. X and R charts have been maintained on a certain process with a
subgroup size of 4. Past data indicate an  of 20.0 and a  of 2.0 .
A suggestion is made to increase the sample size from 4 to 5 but to
maintain the same limits as before on the  and R charts so that “we
shall not appear to be tightening up on the process”. Discuss the
consequences that you would expect if this suggest were followed. If the
suggestion is adopted, would points be more likely or less likely than
before to fall outside the control limits on the chart ? On the R chart?
Explain your answers. (Points will be less likely than before to fall outside
control limits on bar chart, and more likely on R chart)
Q.7. You are shown what purports to be a control chart for  on a certain
quality characteristic of a manufactured product. The control chart
contains 50 subgroups. You observe that the  values are close to the
central line on the chart and none are near the 3-sigma limits. In fact,
when you draw one sigma limits (only one-third of the distance from the
central line to the control limits shown), all the points fall within these
narrow limits.
What possible explanations occur to you that might account for an
chart of this type? (Significant improvement in the process or some errors)
Q 8. A textile mill' s development group determines that it is required to
have a fiber which, among other properties, must have a minimum
allowable tensile strength of 1.30 grams in 99% of the fiber used.
Manufacturer A offers to supply the textile mill with the fiber and a contract
is signed.
a) Manufacturer A knows that the standard deviation () of his process is
0.02. What is the lowest possible target tensile strength to ensure that
exactly 99% of the fiber will have at least 1.30 gram required tensile
strength? Assume that statistical control will be maintained and that
the distribution of tensile strength is approximately normal.
b) The quality control group of Manufacturer A then decides that control
charts will be based on a subgroup size of 16. Determine 3 control
limits for  and  charts assuming that the process average is set at
the value computed in part (a) above. (Target tensile strength=1.3465)