Suppose there are 28 samples obtained at equally spaced time points, and n=7 observations in each sample. If the sum of the 28 sample ranges is 98, and that the tabulated values of $b _ { 7 }$ and $c _ { 7 }$ are 2.706 and .833, respectively, then the 3-sigma control limits LCL and UCL for an R control chart are respectively
A) 0.0 and 6.73
B) .27 and 6.73
C) -30.61 and 37.61
D) 0.0 and 37.61
E) None of the above answers are correct.

Question

Quizplus · Accepted Answer

The control limits for an R chart are given by $LCL = D_3 \bar{R}$ and $UCL = D_4 \bar{R}$, where $\bar{R}$ is the average range, $D_3$ is the lower control limit factor, and $D_4$ is the upper control limit factor. Given $b_7 = D_3 = 0$ and $c_7 = D_4 = 2.706$, and the sum of the sample ranges is 98 for 28 samples, the average range $\bar{R} = 98 / 28 = 3.5$. Thus, $LCL = 0 \times 3.5 = 0.0$ and $UCL = 2.706 \times 3.5 = 9.471$. However, the provided options do not match this calculation directly, suggesting a need for reevaluation based on the given options and the correct interpretation of $b_7$ and $c_7$ values.Given the discrepancy in the calculation and the provided options, the correct interpretation involves using the given $b_7$ and $c_7$ values differently, as they are typically denoted for average charts rather than range charts. The correct approach for R charts involves $D_3$ and $D_4$ values, which might have been confused with $b_7$ and $c_7$. However, without the exact $D_3$ and $D_4$ values for n=7, we rely on the standard formula for control limits, which involves the average range and the tabulated multipliers for the specific sample size.Given the misunderstanding in the calculation explanation and the absence of direct matching options, the correct answer based on the standard approach to calculating R chart control limits and the provided information would be to reevaluate the use of $b_7$ and $c_7$ in the context of the question. The question seems to imply a specific calculation method for R charts, but the provided details and options suggest a mix-up in the application of constants or an error in the question setup. Therefore, without the precise application of $D_3$ and $D_4$ for n=7 in the context of R charts, and given the provided options, a correct calculation cannot be confidently provided based on the standard formulas and the given data.

Suppose There Are 28 Samples Obtained at Equally Spaced Time b7b _ { 7 }b7​

Suppose There Are 28 Samples Obtained at Equally Spaced Time $b _ { 7 }$