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VIPSolutions · Accepted Answer

## Step-by-Step Solution

Let's solve the problem step by step.

### **Given Data**
- Proportion of nonconforming items ($ \overline{p} $) = 17% = **.17**
- Two cases:
  1. **Batch size ($ n $) = 50**
  2. **Batch size ($ n $) = 100**

For a **p-chart**, we use the following formulas:

- **Centerline (CL):**
  $$
  CL = \overline{p}
  $$

- **Standard deviation ($ \sigma_p $):**
  $$
  \sigma_p = \sqrt{\frac{\overline{p}(1-\overline{p})}{n}}
  $$

- **Upper Control Limit (UCL):**
  $$
  UCL = \overline{p} + 3\sigma_p
  $$

- **Lower Control Limit (LCL):**
  $$
  LCL = \overline{p} - 3\sigma_p
  $$
  If LCL is negative, set it to .

---

## **Case 1: Batch Size $ n = 50 $**

### 1. **Calculate the Centerline**
$$
CL = .17
$$

### 2. **Calculate the Standard Deviation**
$$
\sigma_p = \sqrt{\frac{.17 \times (1 - .17)}{50}} = \sqrt{\frac{.17 \times .83}{50}}
$$
$$
= \sqrt{\frac{.1411}{50}} = \sqrt{.002822} \approx .0532
$$

### 3. **Calculate UCL and LCL**
$$
UCL = .17 + 3 \times .0532 = .17 + .1596 = .3296
$$
$$
LCL = .17 - 3 \times .0532 = .17 - .1596 = .0104
$$

---

## **Case 2: Batch Size $ n = 100 $**

### 1. **Calculate the Centerline**
$$
CL = .17
$$

### 2. **Calculate the Standard Deviation**
$$
\sigma_p = \sqrt{\frac{.17 \times (1 - .17)}{100}} = \sqrt{\frac{.1411}{100}} = \sqrt{.001411} \approx .0376
$$

### 3. **Calculate UCL and LCL**
$$
UCL = .17 + 3 \times .0376 = .17 + .1128 = .2828
$$
$$
LCL = .17 - 3 \times .0376 = .17 - .1128 = .0572
$$

---

## **Final Answers (Rounded to 4 decimal places):**

### **A. For batch size = 50:**
- **Centerline (CL):** .170
- **UCL:** .3296
- **LCL:** .0104

### **B. For batch size = 100:**
- **Centerline (CL):** .170
- **UCL:** .2828
- **LCL:** .0572

---

### **Summary Table**

| Batch Size | Centerline (CL) | UCL    | LCL    |
|------------|-----------------|--------|--------|
| 50         | .170          | .3296 | .0104 |
| 100        | .170          | .2828 | .0572 |

If you need further explanation, feel free to ask!

VIPSolutions · Answer

# Step-by-Step Solution

Let's solve the problem step by step.

## Given Data
- **Proportion of nonconforming items ($ \overline{p} $)**: 17% or **0.17**
- **Batch sizes**:
  1. $ n = 50 $
  2. $ n = 100 $

## Formulas for p-chart
- **Centerline (CL)**:
  $$
  CL = \overline{p}
  $$

- **Standard deviation ($ \sigma_p $)**:
  $$
  \sigma_p = \sqrt{\frac{\overline{p}(1-\overline{p})}{n}}
  $$

- **Upper Control Limit (UCL)**:
  $$
  UCL = \overline{p} + 3\sigma_p
  $$

- **Lower Control Limit (LCL)**:
  $$
  LCL = \overline{p} - 3\sigma_p
  $$
  If $ LCL $ is negative, set it to 0.

---

## Case 1: Batch Size $ n = 50 $

### 1. Calculate the Centerline
$$
CL = 0.17
$$

### 2. Calculate the Standard Deviation
$$
\sigma_p = \sqrt{\frac{0.17 \times (1 - 0.17)}{50}} = \sqrt{\frac{0.17 \times 0.83}{50}} = \sqrt{\frac{0.1411}{50}} \approx 0.0532
$$

### 3. Calculate UCL and LCL
$$
UCL = 0.17 + 3 \times 0.0532 = 0.17 + 0.1596 = 0.3296
$$
$$
LCL = 0.17 - 3 \times 0.0532 = 0.17 - 0.1596 = 0.0104
$$

---

## Case 2: Batch Size $ n = 100 $

### 1. Calculate the Centerline
$$
CL = 0.17
$$

### 2. Calculate the Standard Deviation
$$
\sigma_p = \sqrt{\frac{0.17 \times (1 - 0.17)}{100}} = \sqrt{\frac{0.1411}{100}} \approx 0.0376
$$

### 3. Calculate UCL and LCL
$$
UCL = 0.17 + 3 \times 0.0376 = 0.17 + 0.1128 = 0.2828
$$
$$
LCL = 0.17 - 3 \times 0.0376 = 0.17 - 0.1128 = 0.0572
$$

---

## Final Answers (Rounded to 4 decimal places)

### A. For Batch Size $ n = 50 $:
- **Centerline (CL)**: 0.1700
- **UCL**: 0.3296
- **LCL**: 0.0104

### B. For Batch Size $ n = 100 $:
- **Centerline (CL)**: 0.1700
- **UCL**: 0.2828
- **LCL**: 0.0572

---

## Summary Table

| Batch Size | Centerline (CL) | UCL    | LCL    |
|------------|-----------------|--------|--------|
| 50         | 0.1700          | 0.3296 | 0.0104 |
| 100        | 0.1700          | 0.2828 | 0.0572 |

If you have any further questions, feel free to ask!

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