https://file.aiquickdraw.com/mathbot/file/1730277002144zfddg0ck.jpg

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.

Learn to manually calculate the average sample fraction nonconforming (\bar{p}) for quality control in a dataset of orange juice concentrate cans. The solution demonstrates key concepts in statistics and quality control, such as proportion calculations for nonconforming items, suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation