Question

1. An operator trainee is attempting to monitor a filling process that has an overall average of 650 cc. The average range is 22 cc. If you use a sample size of 8, what are the upper and lower control limits for the x-bar chart and R-chart?

A2 = 0.373, D4 = 1.864, D3 = 0.136

2. A small, independent amusement park collects data on the number of cars with out-of-state license plates. The sample size is fixed at n=100 each day. Data from the previous 10 days indicate the following number of out-of-state license plates:

Day Out-of-state Plates
1 6
2 4
3 5
4 7
5 8
6 3
7 4
8 15
9 3
10 11

(a) Calculate the overall proportion of "tourists" (cars with out-of-state plates) and the standard deviation of proportions.
(b) Using ±3σ limits, calculate the LCL and UCL for these data.
(c) Is the process under control? Explain.

Answer 1

1. Xbar = 650 cc

Rbar = 22 cc

Sample size n=8

A₂=0.373 (for n=8)

D₃=0.136 (for n=8)

D₄=1.864 (for n=8)

Control limits for 3 Sigma Xbar Chart

$UCL_x = Xbar + A_2 \times Rbar = 650 + 0.373 \times 22 = 650+8.206 \\ UCL_x = 658.206 \\ LCL_x = Xbar - A_2 \times Rbar = 650 - 0.373 \times 22 = 650 - 8.206 \\ UCL_x = 641.794$

Control limits for 3 Sigma R Chart

$UCLR = D_4 \times Rbar = 1.864 \times 22 =41.008 \\ UCLR =41.008 \\ LCLR = D_3 \times Rbar = 0.136 \times 22 = 2.992 \\ LCLR = 2.992$

2.

(a) The overall proportion of "tourists":

$P = \frac{0.06+0.04+0.05+0.07+0.08+0.03+0.04+0.15+0.03+0.11}{10} = 0.066$

Standard deviation of proportions:

$σ = \sqrt{\frac{0.066 \times (1-0.066)}{100}} = 0.0248$

(b)

$LCL = P - 3σ \\ LCL = 0.066 -3 \times 0.0248 = 0 \\ UCL = P + 3σ \\ UCL = 0.066 + 3 \times 0.0248 = 0.1404$

(c) The process is under control because the proportion of tourists who drive car with out of state plates is within the upper and lower control limits.