Binomial Test Sample Size Calculator

Sample Size Formula for Binomial Proportions:

\[ n = \frac{[Z_{\alpha/2} + Z_{\beta}]^2 \times (p_1(1-p_1) + p_2(1-p_2))}{(p_1 - p_2)^2} \]

Proportion 1 (p₁):

(0-1)

Proportion 2 (p₂):

(0-1)

Alpha (α):

Beta (β):

Required Sample Size (n):

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1. What is Binomial Test Sample Size Calculation?

The binomial test sample size calculation determines the number of participants needed in each group to detect a statistically significant difference between two proportions with specified power and significance level.

2. How Does the Calculator Work?

The calculator uses the standard formula for comparing two proportions:

\[ n = \frac{[Z_{\alpha/2} + Z_{\beta}]^2 \times (p_1(1-p_1) + p_2(1-p_2))}{(p_1 - p_2)^2} \]

Where:

\( n \) — Sample size per group
\( Z_{\alpha/2} \) — Z-score for significance level (two-tailed)
\( Z_{\beta} \) — Z-score for power (one-tailed)
\( p_1 \) — Proportion in group 1
\( p_2 \) — Proportion in group 2

Explanation: This formula calculates the minimum sample size needed to detect a specified difference between two proportions with given statistical power and significance level.

3. Importance of Sample Size Calculation

Details: Proper sample size calculation ensures studies have adequate power to detect meaningful effects, prevents wasted resources on underpowered studies, and helps in research planning and ethical considerations.

4. Using the Calculator

Tips: Enter proportions as decimals between 0 and 1. Select appropriate alpha (significance level) and beta (power) levels based on your study requirements. The calculator provides sample size per group.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between one-tailed and two-tailed tests?
A: Two-tailed tests detect differences in either direction, while one-tailed tests detect differences in one specific direction. This calculator uses two-tailed Z-scores for alpha.

Q2: How do I choose the right alpha and beta values?
A: Alpha=0.05 (95% confidence) and beta=0.20 (80% power) are common standards. Use more stringent values (alpha=0.01, beta=0.10) for higher confidence and power.

Q3: What if my proportions are very close?
A: Smaller differences between proportions require larger sample sizes to detect statistical significance.

Q4: Can this be used for single proportion tests?
A: No, this calculator is specifically for comparing two proportions. Different formulas are used for single proportion tests.

Q5: What about continuity corrections?
A: This formula provides the basic calculation. For small sample sizes or extreme proportions, continuity corrections may be applied for more accurate results.