Estimating Confidence Range

Solution & Explanation

When faced with the problem of estimating a confidence interval for the proportion of defective items in a batch, we can utilize the properties of the binomial distribution. Below is a step-by-step solution and explanation for calculating the confidence interval in this scenario:

Step 1: Understand the Problem

• Given:
  • Total items (n): 100
  • Defective items: 25
• Objective: Calculate the 95% confidence interval for the proportion of defective items.

Step 2: Identify the Proportion

• Sample Proportion ($\hat{p}$): $\hat{p} = \frac{\text{Number of Defective Items}}{\text{Total Items}} = \frac{25}{100} = 0.25$

Step 3: Calculate the Standard Error (SE)

The standard error for a proportion is calculated using the formula:

• Standard Error (SE): $SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.25 \times (1 - 0.25)}{100}} = \sqrt{\frac{0.25 \times 0.75}{100}} = 0.0433$

Step 4: Determine the Z-Score for the Desired Confidence Level

• 95% Confidence Level:
  • The Z-score for a 95% confidence level is approximately 1.96.

Step 5: Calculate the Margin of Error (ME)

• Margin of Error (ME): $ME = Z \times SE = 1.96 \times 0.0433 = 0.0848$

Step 6: Calculate the Confidence Interval (CI)

• Confidence Interval (CI): $CI = \hat{p} \pm ME = 0.25 \pm 0.0848$
  • Lower Bound: $0.25 - 0.0848 = 0.1652$
  • Upper Bound: $0.25 + 0.0848 = 0.3348$

Final Result

• 95% Confidence Interval for the Proportion of Defective Items: $\text{CI} = [0.1652, 0.3348]$

Explanation

• Interpretation:
  • We are 95% confident that the true proportion of defective items in the entire population lies between 16.52% and 33.48%.
• Assumptions:
  • The sample is randomly selected.
  • The sample size is large enough to assume a normal approximation of the binomial distribution (n$\hat{p}$ and n(1-$\hat{p}$ are both greater than 5).
• Importance:
  • Confidence intervals provide a range of plausible values for the population parameter, offering a more comprehensive insight than a single point estimate.