To manually calculate the average sample fraction nonconforming $\bar{p}$ for the data given, we can follow these steps:

1. Identify the number of nonconforming cans $D_i$ and sample size $n = 50$.
2. Calculate each sample fraction nonconforming $\hat{p}_i = \frac{D_i}{n}$ for each sample number.
3. Sum all $\hat{p}_i$ values and divide by the total number of samples (24 in this case) to get $\bar{p}$.

The data provided already lists $\hat{p}_i$ values for each sample, and the total of nonconforming cans $D$ across all samples is given as 133. Here's how the calculation of $\bar{p}$ would proceed:

$\bar{p} = \frac{\sum \hat{p}_i}{24} = \frac{133}{24 \times 50} = \frac{133}{1200} = 0.1108$

Thus, $\bar{p} = 0.1108$ matches the result already given in the table.

Would you like further details or have any questions on this?

Here are some related questions to extend your understanding:

1. What does the sample fraction nonconforming $\hat{p}_i$ represent in quality control?
2. How is the value of $\bar{p}$ useful in control chart analysis?
3. How would you calculate the standard deviation of $\hat{p}_i$ values?
4. Why is it important to use a large sample size $n$ in quality control?
5. How could outliers in $\hat{p}_i$ values impact the interpretation of $\bar{p}$?

Tip: For quality control, always ensure sample data is collected consistently to maintain accurate $\bar{p}$ values and other statistical measurements.