Question

# A branch manager of a brokerage company is concerned with the number of undesirable trades made by her sales staff. A trade is considered undesirable if there is an error on the trade ticket. Trades with errors are canceled and resubmitted. The cost of correcting errors is billed to the brokerage company. The branch manager wants to know whether the proportion of undesirable trades is in a state of statistical control so she can plan the next step in a quality improvement process. Data were collected for a 30 -day period and stored in Trade.a. Construct a control chart for these data.b. Is the process in control? Explain.c. Based on the results of (a) and (b), what should the manager do next to improve the process?

Solution

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We are tasked to construct a $p$ chart from the given data set of Errors. A $\color{#4257b2}\textbf{p-chart}$ is a type of attribute control chart that is used to monitor processes with categorical variables. It is used to detect $\color{#4257b2}\textbf{special cause variations}$, or the fluctuations that goes outside the control limits of a process. The "$p$" derives its name from the word proportion, which are the values we'll be working closely with in constructing the chart.

Furthermore, we shall be determining the state of control of the process by observing the chart generated and look for points that occurs outside the set control limits. These control limits are called the $\color{#4257b2}\textbf{Upper Control Limit (UCL)}$ and the $\color{#4257b2}\textbf{Lower Control Limit (LCL)}$, and are given by Eqs. $(1)$ and $(2)$.

\begin{align} UCL &= \bar p+3\sqrt{\dfrac{\bar p(1-\bar p)}{\bar n}}\\ LCL &= \bar p-3\sqrt{\dfrac{\bar p(1-\bar p)}{\bar n}}\\ \bar n&=\sum^k_{i=1}n_i\dfrac{1}{k}\\ \bar p&=\sum^k_{i=1}X_i\dfrac{1}{\sum^k_{i=1}n_i} \end{align}

where:

• $\bar p$ is the proportion of nonconforming items in the $k$ subgroups combines
• $k$ number of subgroups selected
• $\bar n$ mean subgroup size
• $X_i$ is the number of nonconforming items in subgroup $i$
• $n_i$ is the sample size for subgroup $i$

For this problem, we are going to:

1. Construct a $p$-chart using the given data and plot the $UCL$ and $LCL$ values.
2. Observe if a point exists outside the control limits. If it does, then consider the process as not in a state of control.
3. In the case of out-of-control processes, remove the outlier and recalculate the control limits.

What does it mean when we say that the process is not in control?