🧮 Determination of Confidence Intervals

A Confidence Interval (CI) gives a range of values within which the true population parameter (mean, proportion, risk ratio, etc.) is expected to fall — with a specified level of confidence (e.g., 95%).

🔹 Step-by-Step Process

Step 1: Collect Data

Obtain sample data — for example:

• Mean or proportion

• Standard deviation (SD)

• Sample size (n)

Step 2: Choose the Confidence Level

Decide how confident you want to be:

• 90% → Z = 1.645

• 95% → Z = 1.96

• 99% → Z = 2.576

Higher confidence = wider interval (more certain, less precise).

Step 3: Calculate the Standard Error (SE)

For Mean:

SE=SDnSE = \frac{SD}{\sqrt{n}}SE=n​SD​

For Proportion:

SE=p(1−p)nSE = \sqrt{\frac{p(1-p)}{n}}SE=np(1−p)​​

Where:

• p = sample proportion (e.g., 0.45 for 45%)

Step 4: Determine the Margin of Error (ME)

$$ME=Z\times SE$$

Step 5: Construct the Confidence Interval

Confidence Interval=Xˉ±ME\text{Confidence Interval} = \bar{X} \pm MEConfidence Interval=Xˉ±ME

or for proportions:

$$CI=p\pm Z\times SE$$

🔹 Example 1: Mean

A study on systolic BP (mmHg):

• Mean (X̄) = 130

• SD = 10

• n = 25

• Confidence Level = 95%

Step 1: SE = 10 / √25 = 10 / 5 = 2
Step 2: ME = 1.96 × 2 = 3.92
Step 3: CI = 130 ± 3.92 → (126.08 – 133.92 mmHg)

✅ Interpretation: We are 95% confident the true population mean lies between 126.08 and 133.92 mmHg.

🔹 Example 2: Proportion

In a vaccine study:

• p = 0.80 (80% effectiveness)

• n = 100

• Confidence Level = 95%

Step 1: SE = √[0.8(1−0.8)/100] = √(0.16/100) = 0.04
Step 2: ME = 1.96 × 0.04 = 0.078
Step 3: CI = 0.80 ± 0.078 = (0.722 – 0.878) or 72.2%–87.8%

🔹 Key Insights

🔹 Clinical Relevance

Confidence intervals are used to:

• Evaluate precision of study results

• Check statistical significance (if CI excludes 0 or 1)

• Compare treatment effects between groups