Questions: For a process such as this to be "in statistical control," only random patterns of points should appear on the control chart. The process is "not in statistical control" when any non-random patterns appear on the chart. Three commonly used tests or rules for signaling that a process may not be in statistical control are: - Rule 1: One or more points fall outside one of the chart's control limits - Rule 2: Nine points in a row are on the same side of the chart's centerline - Rule 3: Six points in a row are increasing (or decreasing). Using these three rules, identify which (if any) of the rules gives an "out of control" signal, and indicate the subgroup number at which the signal is first given.

For a process such as this to be "in statistical control," only random patterns of points should appear on the control chart. The process is "not in statistical control" when any non-random patterns appear on the chart. Three commonly used tests or rules for signaling that a process may not be in statistical control are:
- Rule 1: One or more points fall outside one of the chart's control limits
- Rule 2: Nine points in a row are on the same side of the chart's centerline
- Rule 3: Six points in a row are increasing (or decreasing).

Using these three rules, identify which (if any) of the rules gives an "out of control" signal, and indicate the subgroup number at which the signal is first given.

Transcript text: For a process such as this to be "in statistical control," only random patterns of points should appear on the control chart. The process is "not in statistical control" when any non-random patterns appear on the chart. Three commonly used tests or rules for signaling that a process may not be in statistical control are: - Rule 1: One or more points fall outside one of the chart's control limits - Rule 2: Nine points in a row are on the same side of the chart's centerline - Rule 3: Six points in a row are increasing (or decreasing). Using these three rules, identify which (if any) of the rules gives an "out of control" signal, and indicate the subgroup number at which the signal is first given.

Solution

Solution Steps

To determine if the process is "out of control" based on the given rules, we need to analyze the data points on the control chart. We will check for the following conditions:

If any point falls outside the control limits.
If there are nine consecutive points on the same side of the centerline.
If there are six consecutive points that are either increasing or decreasing.

Step 1: Check Rule 1 - Points Outside Control Limits

We need to determine if any point in the data set falls outside the control limits. The control limits are given as:

Upper control limit: \( UCL = 25 \)
Lower control limit: \( LCL = 5 \)

From the data set: \[ \{10, 12, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24\} \]

None of the points fall outside the control limits. Therefore, Rule 1 does not signal an "out of control" process.

Step 2: Check Rule 2 - Nine Consecutive Points on the Same Side of the Centerline

The centerline is calculated as: \[ \text{Centerline} = \frac{UCL + LCL}{2} = \frac{25 + 5}{2} = 15 \]

We need to check if there are nine consecutive points either all above or all below the centerline. From the data set, starting from the 7th point (index 6): \[ \{16, 17, 18, 19, 20, 21, 22, 23, 24\} \]

All these points are above the centerline (\(15\)). Therefore, Rule 2 signals an "out of control" process at the 15th point.

Step 3: Check Rule 3 - Six Consecutive Points Increasing or Decreasing

We need to check if there are six consecutive points that are either increasing or decreasing. From the data set, starting from the 3rd point (index 2): \[ \{11, 13, 14, 15, 16, 17, 18\} \]

These points are in an increasing sequence. Therefore, Rule 3 signals an "out of control" process at the 8th point.