The sample size calculation was based on the study conducted by Goel et al in the health facilities in North India (7). The reported patient satisfaction was 87.8%. Considering this proportion, with absolute precision of 5% at 95% confidence interval, we calcu lated a sample size of 187 which was rounded off to 200.
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This is a biostatistics/research methodology question about sample size calculation. Let me verify the math behind it.
The formula used here is for estimating a single proportion:
$$n = \frac{Z^2 \cdot p(1-p)}{d^2}$$
Where:
- Z = 1.96 (for 95% confidence interval)
- p = 0.878 (reported patient satisfaction of 87.8%)
- d = 0.05 (absolute precision of 5%)
Step-by-step calculation:
1. p(1 − p):
0.878 × (1 − 0.878) = 0.878 × 0.122 = 0.107116
2. Z²:
1.96² = 3.8416
3. d²:
0.05² = 0.0025
4. n:
n = (3.8416 × 0.107116) / 0.0025
n = 0.41149 / 0.0025
n = 164.6 ≈ 165
Discrepancy with the stated 187:
The calculated value using standard inputs gives ~165, not 187. The figure of 187 would result if a slightly different value of p or Z were used. For example:
- If p = 0.90 is used instead: 3.8416 × 0.09 / 0.0025 = 138 — still not 187.
- If Z = 2.05 (97.5% CI, two-tailed correction) were used: not standard.
- If the authors used p = 0.878 with a finite population correction or a design effect (DEFF), the number could be inflated to ~187 before rounding to 200.
A more likely explanation: they may have applied a 10–15% non-response/attrition adjustment:
165 × 1.134 ≈ 187 ✓
This is a common practice — inflating the sample size to account for non-response, dropout, or incomplete data. So the final sequence would be:
Raw n = 165 → adjusted for ~13% non-response → 187 → rounded to 200
Summary Table:
| Parameter | Value Used |
|---|---|
| Reference proportion (p) | 87.8% |
| Absolute precision (d) | 5% |
| Confidence level | 95% (Z = 1.96) |
| Calculated n (raw) | ~165 |
| Adjusted n (non-response) | ~187 |
| Final sample size | 200 |
The core formula and rationale are statistically sound. The gap between 165 and 187 is most plausibly explained by a non-response rate adjustment (~13%), which the authors did not explicitly state in the excerpt you shared.
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