Construct the appropriate control chart for the observations listed below, using 3 sigma limits.

Product Number - Number of defects
1 - 3
2 - 3
3 - 6
4 - 5
5 - 3
6 - 5
7 - 2
8 - 2
9 - 3
10 - 4
11 - 6

a. Determine the upper and lower control limits.

b. Discuss the chart you have plotted. What conclusions and recommendations do you make from the chart and the data?

Here's what you will need to answer this question:

Lower limits = mean - 3(sd/√n)
Upper limits = mean + 3(sd/√n)

sd = standard deviation
n = sample size

You will need to calculate the mean and standard deviation from the data given. Once you have those values, you can then determine the lower and upper limits.

I hope this will help get you started.

To construct the control chart, we'll first calculate the mean and standard deviation of the observations.

The given data is as follows:
Product Number - Number of defects
1 - 3
2 - 3
3 - 6
4 - 5
5 - 3
6 - 5
7 - 2
8 - 2
9 - 3
10 - 4
11 - 6

Step 1: Calculate the mean (x̄) of the observations.
Mean (x̄) = (3 + 3 + 6 + 5 + 3 + 5 + 2 + 2 + 3 + 4 + 6) / 11 = 3.909

Step 2: Calculate the standard deviation (σ) of the observations.
To calculate the standard deviation, we need the variance. The variance (σ²) is calculated using the formula:
Variance (σ²) = Σ(x - x̄)² / N, where Σ represents summation, x is each observation, x̄ is the mean, and N is the number of observations.

Using the given data, we have:
Variance = ((3 - 3.909)² + (3 - 3.909)² + (6 - 3.909)² + (5 - 3.909)² + (3 - 3.909)² + (5 - 3.909)² + (2 - 3.909)² + (2 - 3.909)² + (3 - 3.909)² + (4 - 3.909)² + (6 - 3.909)²) / 11
Variance = 2.737

Standard Deviation (σ) = √(Variance) = √(2.737) ≈ 1.653

Step 3: Calculate the 3-sigma limits.
Upper Control Limit (UCL) = x̄ + (3 * σ)
Lower Control Limit (LCL) = x̄ - (3 * σ)

UCL = 3.909 + (3 * 1.653) = 8.868
LCL = 3.909 - (3 * 1.653) = -1.05

a. The upper control limit (UCL) is approximately 8.868, and the lower control limit (LCL) is approximately -1.05.

b. The control chart is a graph that displays the observations (number of defects) over time. The central line represents the mean, and the upper and lower control limits represent the expected variation around the mean.

From the chart and the data, we can observe the following conclusions and make recommendations:
1. None of the observations fall outside the 3-sigma limits, indicating that the process is within statistical control.
2. The number of defects varies around the mean of 3.909.
3. The observations seem to exhibit a stable behavior as they fall within the control limits.
4. The process should be monitored regularly to ensure it remains within control.
5. If any observations fall outside the control limits in the future, it may indicate a special cause of variation that needs to be investigated and addressed.

It is important to note that control charts are most effective when used in conjunction with other statistical process control techniques and quality improvement methods to identify and address the root causes of variation and improve the overall process performance.