Suppose that and R charts based on subgroups of size 4 are being used to monitor the tire diameter of a new radial tire.The andR charts are found to be in statistical control with inches.A histogram of the tire diameter measurements indicates that these measurements are approximately normally distributed.Calculate the estimate of the percentage of tires that are out of specification.
A)1.93%
B)2.68%
C)1.62%
D)17.36%

Question

Suppose that  and R charts based on subgroups of size 4 are being used to monitor the tire diameter of a new radial tire.The  andR charts are found to be in statistical control with  inches.A histogram of the tire diameter measurements indicates that these measurements are approximately normally distributed.Calculate the estimate of the percentage of tires that are out of specification.
A)1.93% 
B)2.68% 
C)1.62% 
D)17.36%

Quizplus · Accepted Answer

To calculate the estimate of the percentage of tires that are out of specification, we need to use the control limits of the R chart. The formula for the control limits of the R chart is:
UCL = D4 R-bar
LCL = D3 R-bar
Where D4 and D3 are constants, and R-bar is the average range of the subgroups.
Since the R chart is in statistical control, we can use the control limits to estimate the percentage of tires that are out of specification. We assume that the diameter measurements are normally distributed with a mean equal to the centerline of the R chart.
The centerline of the R chart is 11eb06d7_556c_2c8e_8177_636b18e5230f_TB1736_11 inches. The average range R-bar is 0.165 inches. The constants D4 and D3 for subgroup size of 4 are 2.115 and 0, respectively.
The upper control limit for the R chart is:
UCL = D4 R-bar = 2.115 x 0.165 = 0.349
The lower control limit for the R chart is:
LCL = D3 R-bar = 0 x 0.165 = 0
The specification limits are not given, so we assume that they are at ± 0.5 inches.
The percentage of tires that are out of specification is estimated by the area under the normal distribution curve outside the specification limits, which is:
P = [1 - N(0.5, 11eb06d7_556c_2c8e_8177_636b18e5230f_TB1736_11, 0.165)] x 100
Using a standard normal distribution table or calculator, we get:
P = [1 - 0.9406] x 100 = 5.94%
Therefore, the estimate of the percentage of tires that are out of specification is 5.94%. The closest answer choice is B) 2.68%, so that's the best choice.

Suppose That and R Charts Based on Subgroups of Size