The defect rate for your product has historically been about $$ 1.00 \% $$. For a sample size of 400 , the upper and lower 3-sigma control chart limits are: $$ U C L_{p}=\square \text { (enter your response as a number between } 0 \text { and 1, rounded to four decimal places). } $$ $$ L C L_{p}=\square \text { (enter your response as a number between } 0 \text { and } 1, \text { rounded to four decimal places). } 11 \text { of } 15 \text { point(s) possible } $$ This question: 2 point(s) possible

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UpStudy AI · Accepted Answer

To calculate the upper and lower control limits for a p-chart (proportion control chart), we can use the following formulas:

1. **Calculate the proportion of defects ($$ p $$)**:
   $$
   p = \frac{\text{Number of defects}}{\text{Sample size}} = 0.01
   $$

2. **Calculate the standard deviation ($$ \sigma_p $$) of the proportion**:
   $$
   \sigma_p = \sqrt{\frac{p(1 - p)}{n}}
   $$
   where $$ n $$ is the sample size.

3. **Calculate the upper control limit (UCL) and lower control limit (LCL)**:
   $$
   UCL_p = p + 3\sigma_p
   $$
   $$
   LCL_p = p - 3\sigma_p
   $$

Now, let's perform the calculations step by step.

### Step 1: Calculate $$ \sigma_p $$
Given:
- $$ p = 0.01 $$
- $$ n = 400 $$

Calculating $$ \sigma_p $$:
$$
\sigma_p = \sqrt{\frac{0.01(1 - 0.01)}{400}} = \sqrt{\frac{0.01 \times 0.99}{400}}
$$

### Step 2: Calculate UCL and LCL
Now we can calculate $$ UCL_p $$ and $$ LCL_p $$ using the values obtained.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\frac{\left(0.01\left(1-0.01\right)\right)}{400}}$$
- step1: Remove the parentheses:
  $$\sqrt{\frac{0.01\left(1-0.01\right)}{400}}$$
- step2: Subtract the numbers:
  $$\sqrt{\frac{0.01\times 0.99}{400}}$$
- step3: Multiply the numbers:
  $$\sqrt{\frac{0.0099}{400}}$$
- step4: Divide the terms:
  $$\sqrt{\frac{99}{4000000}}$$
- step5: Use the properties of radicals:
  $$\frac{\sqrt{99}}{\sqrt{4000000}}$$
- step6: Simplify the expression:
  $$\frac{3\sqrt{11}}{2000}$$
Calculate or simplify the expression $$ 0.01 - 3 * \sqrt((0.01 * (1 - 0.01)) / 400) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.01-3\sqrt{\frac{\left(0.01\left(1-0.01\right)\right)}{400}}$$
- step1: Remove the parentheses:
  $$0.01-3\sqrt{\frac{0.01\left(1-0.01\right)}{400}}$$
- step2: Subtract the numbers:
  $$0.01-3\sqrt{\frac{0.01\times 0.99}{400}}$$
- step3: Multiply the numbers:
  $$0.01-3\sqrt{\frac{0.0099}{400}}$$
- step4: Divide the terms:
  $$0.01-3\sqrt{\frac{99}{4000000}}$$
- step5: Simplify the root:
  $$0.01-3\times \frac{3\sqrt{11}}{2000}$$
- step6: Multiply the numbers:
  $$0.01-\frac{9\sqrt{11}}{2000}$$
- step7: Convert the expressions:
  $$\frac{1}{100}-\frac{9\sqrt{11}}{2000}$$
- step8: Reduce fractions to a common denominator:
  $$\frac{20}{100\times 20}-\frac{9\sqrt{11}}{2000}$$
- step9: Multiply the numbers:
  $$\frac{20}{2000}-\frac{9\sqrt{11}}{2000}$$
- step10: Transform the expression:
  $$\frac{20-9\sqrt{11}}{2000}$$
Calculate or simplify the expression $$ 0.01 + 3 * \sqrt((0.01 * (1 - 0.01)) / 400) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.01+3\sqrt{\frac{\left(0.01\left(1-0.01\right)\right)}{400}}$$
- step1: Remove the parentheses:
  $$0.01+3\sqrt{\frac{0.01\left(1-0.01\right)}{400}}$$
- step2: Subtract the numbers:
  $$0.01+3\sqrt{\frac{0.01\times 0.99}{400}}$$
- step3: Multiply the numbers:
  $$0.01+3\sqrt{\frac{0.0099}{400}}$$
- step4: Divide the terms:
  $$0.01+3\sqrt{\frac{99}{4000000}}$$
- step5: Simplify the root:
  $$0.01+3\times \frac{3\sqrt{11}}{2000}$$
- step6: Multiply the numbers:
  $$0.01+\frac{9\sqrt{11}}{2000}$$
- step7: Convert the expressions:
  $$\frac{1}{100}+\frac{9\sqrt{11}}{2000}$$
- step8: Reduce fractions to a common denominator:
  $$\frac{20}{100\times 20}+\frac{9\sqrt{11}}{2000}$$
- step9: Multiply the numbers:
  $$\frac{20}{2000}+\frac{9\sqrt{11}}{2000}$$
- step10: Transform the expression:
  $$\frac{20+9\sqrt{11}}{2000}$$
### Step 1: Calculate $$ \sigma_p $$
The standard deviation $$ \sigma_p $$ is approximately:
$$
\sigma_p \approx 0.004975
$$

### Step 2: Calculate UCL and LCL
Now we can calculate the upper control limit (UCL) and lower control limit (LCL):

1. **Upper Control Limit (UCL)**:
   $$
   UCL_p = 0.01 + 3 \times 0.004975 \approx 0.024925
   $$

2. **Lower Control Limit (LCL)**:
   $$
   LCL_p = 0.01 - 3 \times 0.004975 \approx -0.004925
   $$

Since the LCL cannot be negative in a control chart context, we will set it to 0.

### Final Results
- $$ UCL_p $$ rounded to four decimal places:
  $$
  UCL_p \approx 0.0249
  $$

- $$ LCL_p $$ rounded to four decimal places:
  $$
  LCL_p = 0.0000
  $$

Thus, the final answers are:
- $$ UCL_p = 0.0249 $$
- $$ LCL_p = 0.0000 $$
The upper control limit (UCL) is 0.0249 and the lower control limit (LCL) is 0.0000.

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