To construct a control chart using 98% limits for the given problem, let's follow these steps in detail:
### Step-by-Step Solution
Step 1: Determine the proportions and given parameters
- Average proportion of errors, [tex]\(\bar{p} = 0.03\)[/tex]
- Sample size, [tex]\(n = 100\)[/tex]
- Z-score for 98% control limits, [tex]\(z = 2.33\)[/tex]
Step 2: Calculate the standard error
The standard error for the proportion is given by the formula:
[tex]\[
SE = \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}}
\][/tex]
Plugging in the values:
[tex]\[
SE = \sqrt{\frac{0.03 \times (1 - 0.03)}{100}}
= \sqrt{\frac{0.03 \times 0.97}{100}}
= \sqrt{0.000291}
\approx 0.017
\][/tex]
Step 3: Calculate the Upper Control Limit (UCL)
The UCL is calculated as:
[tex]\[
UCL = \bar{p} + z \times SE
\][/tex]
Substituting [tex]\(\bar{p} = 0.03\)[/tex], [tex]\(z = 2.33\)[/tex], and [tex]\(SE = 0.017\)[/tex]:
[tex]\[
UCL = 0.03 + 2.33 \times 0.017
= 0.03 + 0.03961
\approx 0.07
\][/tex]
Step 4: Calculate the Lower Control Limit (LCL)
The LCL is calculated as:
[tex]\[
LCL = \bar{p} - z \times SE
\][/tex]
Substituting [tex]\(\bar{p} = 0.03\)[/tex], [tex]\(z = 2.33\)[/tex], and [tex]\(SE = 0.017\)[/tex]:
[tex]\[
LCL = 0.03 - 2.33 \times 0.017
= 0.03 - 0.03961
= -0.0096
\][/tex]
Since the LCL cannot be less than 0:
[tex]\[
LCL = \max(0, -0.0096) = 0
\][/tex]
Step 5: Assemble the control limits table
[tex]\[
\begin{tabular}{lll}
& UCL & LCL \\
Control Limits & 0.070 & 0 \\
\hline
\end{tabular}
\][/tex]
### Conclusion
a-1 Control Limits:
[tex]\[
\begin{tabular}{lll}
& UCL & LCL \\
Control Limits & 0.070 & 0 \\
\hline
\end{tabular}
\][/tex]
a-2 Process Control Status:
Based on the manager's assumption of the process error rate (which is [tex]\(\bar{p} = 0.03\)[/tex]), the process appears to be within the control limits (0 and 0.070). Therefore, the process is in control.
Is the process in control?
Yes
