Binomial Test: A statistical test used to determine whether the observed proportion of successes in a binary outcome experiment differs significantly from a hypothesized proportion.
Number of Successes (k): The observed number of successes in the experiment.
Success Probability (p): The hypothesized probability of success.
Sample Size (n): The total number of trials or observations in the experiment.
Significance: The binomial test is particularly useful for small sample sizes and provides a p-value to assess the significance of the observed difference.
Example
Let's say the number of successes (k) is 30 and the success probability (p) is 0.6. Using the formula:
\[
n = \frac{30}{0.6} = 50
\]
So, the Sample Size (n) is 50.
Extended information about "Binomial-Test-Sample-Size-Calculator"
One Sample Binomial Test
Definition: A statistical test used to determine if the proportion of successes in a sample is significantly different from a hypothesized proportion.
Formula: \( P = \frac{k!}{x!(k-x)!} p^x (1-p)^{k-x} \)
\( P \): Probability of x successes
\( k \): Number of trials
\( x \): Number of successes
\( p \): Probability of success
Example: \( P = \frac{10!}{3!(10-3)!} 0.5^3 (1-0.5)^{10-3} \)
\( P \): Probability of 3 successes
\( k \): 10 trials
\( x \): 3 successes
\( p \): 0.5
Binomial Distribution Sample Size
Definition: The sample size required for a binomial distribution to achieve a desired level of accuracy.
Formula: \( n = \frac{Z^2 p (1-p)}{E^2} \)
\( n \): Sample size
\( Z \): Z-value (standard normal distribution)
\( p \): Estimated proportion of success
\( E \): Margin of error
Example: \( n = \frac{1.96^2 \times 0.5 \times (1-0.5)}{0.05^2} \)
\( n \): Sample size
\( Z \): 1.96
\( p \): 0.5
\( E \): 0.05
Negative Binomial Sample Size Calculation
Definition: The sample size required for a negative binomial distribution to achieve a desired level of accuracy.
Formula: \( n = \frac{Z^2 p (1-p)}{E^2} \)
\( n \): Sample size
\( Z \): Z-value (standard normal distribution)
\( p \): Estimated proportion of success
\( E \): Margin of error
Example: \( n = \frac{2.58^2 \times 0.3 \times (1-0.3)}{0.1^2} \)
