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The problem involves estimating the defect rate in a factory with a 95% confidence level based on a sample of products. Here's the breakdown:

### Problem Statement
- A sample of \( n = 400 \) products was selected.
- Out of these, \( x = 80 \) were found to be defective.
- We need to estimate the defect rate (proportion) in the entire production with a 95% confidence level.

### Step 1: Calculate the Sample Proportion
The sample proportion \( \hat{p} \) is calculated as:

\[
\hat{p} = \frac{x}{n} = \frac{80}{400} = 0.2
\]

### Step 2: Find the Standard Error (SE)
The standard error for the proportion is calculated as:

\[
SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.2 \times 0.8}{400}} = \sqrt{\frac{0.16}{400}} = \sqrt{0.0004} = 0.02
\]

### Step 3: Determine the Z-Score for a 95% Confidence Level
For a 95% confidence level, the Z-score is approximately 1.96.

### Step 4: Calculate the Confidence Interval
The confidence interval is calculated as:

\[
\text{Confidence Interval} = \hat{p} \pm Z \times SE
\]

\[
= 0.2 \pm 1.96 \times 0.02
\]

\[
= 0.2 \pm 0.0392
\]

This gives us the interval:

\[
[0.1608, 0.2392]
\]

### Step 5: Convert to Percentage
The defect rate in percentage terms is:

\[
[16.08\%, 23.92\%]
\]

### Final Answer
The defect rate in the entire production, with 95% confidence, is estimated to be between **16.08% and 23.92%**.

Would you like further details or have any questions? Here are some related questions to expand your understanding:

1. What would the confidence interval be with a 99% confidence level?
2. How does increasing the sample size affect the confidence interval?
3. What if the sample proportion was 0.25 instead of 0.2? How would that affect the result?
4. How is the Z-score derived for a given confidence level?
5. What assumptions are made in this confidence interval calculation?
6. How can we interpret the confidence interval in practical terms?
7. How would you calculate the margin of error from this interval?
8. What if the sample size was only 100? How would the confidence interval change?

**Tip:** Confidence intervals provide a range within which we expect the true parameter to lie, but it does not guarantee that the parameter is within that range every time.

Learn how to estimate the defect rate in a factory with 95% confidence using statistical analysis. Explore step-by-step calculations of sample proportion, standard error, and confidence intervals. Understand the application of the Central Limit Theorem in practical scenarios.

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