Part 2 of 2 The defect rate for your product has historically been about $$ 1.00 \% $$. For a sample size of 100, the upper and lower 3-sigma control chart limits are: 8 of $$ 8 \mid $$ My score: $$ 6.83 / 8 $$ pts ( $$ 85.34 \% $$ ) $$ U C L_{p}=0.0399 $$ (enter your response as a number between 0 and 1, rounded to four decimal places). $$ L C L_{p}=\square $$ (enter your response as a number between 0 and 1, rounded to four decimal places).

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UpStudy AI · Accepted Answer

To calculate the lower control limit (LCL) for the defect rate in a control chart, we can use the following formula for the control limits in a p-chart:

$$
UCL_p = p + 3 \sqrt{\frac{p(1-p)}{n}}
$$
$$
LCL_p = p - 3 \sqrt{\frac{p(1-p)}{n}}
$$

Where:
- $$ p $$ is the historical defect rate (as a decimal),
- $$ n $$ is the sample size.

Given:
- Historical defect rate $$ p = 1.00\% = 0.01 $$
- Sample size $$ n = 100 $$

### Step 1: Calculate the standard error

First, we need to calculate the standard error:

$$
SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.01(1-0.01)}{100}}
$$

### Step 2: Calculate the LCL

Now we can calculate the LCL using the formula:

$$
LCL_p = p - 3 \cdot SE
$$

Let's perform these calculations step by step.

1. Calculate the standard error $$ SE $$.
2. Calculate the lower control limit $$ LCL_p $$.

I'll perform these calculations now.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\frac{\left(0.01\left(1-0.01\right)\right)}{100}}$$
- step1: Remove the parentheses:
  $$\sqrt{\frac{0.01\left(1-0.01\right)}{100}}$$
- step2: Subtract the numbers:
  $$\sqrt{\frac{0.01\times 0.99}{100}}$$
- step3: Multiply the numbers:
  $$\sqrt{\frac{0.0099}{100}}$$
- step4: Divide the terms:
  $$\sqrt{\frac{99}{1000000}}$$
- step5: Use the properties of radicals:
  $$\frac{\sqrt{99}}{\sqrt{1000000}}$$
- step6: Simplify the expression:
  $$\frac{3\sqrt{11}}{1000}$$
Calculate or simplify the expression $$ 0.01 - 3*\sqrt((0.01*(1-0.01))/100) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.01-3\sqrt{\frac{\left(0.01\left(1-0.01\right)\right)}{100}}$$
- step1: Remove the parentheses:
  $$0.01-3\sqrt{\frac{0.01\left(1-0.01\right)}{100}}$$
- step2: Subtract the numbers:
  $$0.01-3\sqrt{\frac{0.01\times 0.99}{100}}$$
- step3: Multiply the numbers:
  $$0.01-3\sqrt{\frac{0.0099}{100}}$$
- step4: Divide the terms:
  $$0.01-3\sqrt{\frac{99}{1000000}}$$
- step5: Simplify the root:
  $$0.01-3\times \frac{3\sqrt{11}}{1000}$$
- step6: Multiply the numbers:
  $$0.01-\frac{9\sqrt{11}}{1000}$$
- step7: Convert the expressions:
  $$\frac{1}{100}-\frac{9\sqrt{11}}{1000}$$
- step8: Reduce fractions to a common denominator:
  $$\frac{10}{100\times 10}-\frac{9\sqrt{11}}{1000}$$
- step9: Multiply the numbers:
  $$\frac{10}{1000}-\frac{9\sqrt{11}}{1000}$$
- step10: Transform the expression:
  $$\frac{10-9\sqrt{11}}{1000}$$
### Step 1: Calculate the Standard Error

The standard error $$ SE $$ is calculated as follows:

$$
SE = \sqrt{\frac{0.01(1-0.01)}{100}} \approx 0.00995
$$

### Step 2: Calculate the Lower Control Limit (LCL)

Now, we can calculate the lower control limit $$ LCL_p $$:

$$
LCL_p = 0.01 - 3 \cdot 0.00995 \approx -0.01985
$$

Since control limits cannot be negative, we set the lower control limit to 0.

### Final Answer

Thus, the lower control limit $$ LCL_p $$ is:

$$
LCL_p = 0.0000
$$

So, the final answers are:
- $$ UCL_p = 0.0399 $$
- $$ LCL_p = 0.0000 $$
The lower control limit $$ LCL_p $$ is 0.0000.

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