Calculate probability for coin flips

Question

Given the following scenario about user conversion, calculate the the sample size needed.

Codemia · Accepted Answer

To determine the sample size needed for user conversion analysis, we need to establish the parameters of our hypothesis test. Let's define:
  - **p**: the true conversion rate of users (unknown and needs to be estimated)
  - **p0**: the baseline conversion rate (current rate before the change)
  - **α**: the significance level (commonly set at 0.05)
  - **β**: the power of the test (commonly set at 0.80, meaning we want an 80% chance of detecting an effect if there is one)

The key formulas we'll use are based on the normal approximation of a binomial distribution:

1. **Sample size formula for proportion**: 
   $$ n = \frac{(Z_{1-\alpha/2} + Z_{1-\beta})^2 \cdot (p0(1-p0) + p1(1-p1))}{(p1 - p0)^2} $$
   where:
   - $Z_{1-\alpha/2}$: Z-score corresponding to the confidence level (e.g., 1.96 for 95% confidence)
   - $Z_{1-\beta}$: Z-score for the desired power (e.g., 0.84 for 80% power)
   - $p1$: the expected conversion rate after the intervention (to be defined based on expectations)

We will proceed with the following steps to calculate the sample size:
1. Define the baseline conversion rate $p0$ and the expected conversion rate $p1$.
2. Determine the significance level $α$ and the power $β$.
3. Calculate the Z-scores for the specified confidence level and power.
4. Substitute the values into the sample size formula to solve for $n$.

Assuming:
- Baseline conversion rate, $p0 = 0.1$ (10%)
- Expected conversion rate, $p1 = 0.15$ (15%)
- Significance level, $α = 0.05$
- Power, $β = 0.80$

1. Calculate Z-scores:
   - For $α = 0.05$: $Z_{1-\alpha/2} = 1.96$
   - For $β = 0.20$: $Z_{1-\beta} = 0.84$

2. Calculate $p0(1-p0)$ and $p1(1-p1)$:
   - $p0(1-p0) = 0.1(1-0.1) = 0.09$
   - $p1(1-p1) = 0.15(1-0.15) = 0.1275$

3. Substitute into the sample size formula:
   $$ n = \frac{(1.96 + 0.84)^2 \cdot (0.09 + 0.1275)}{(0.15 - 0.1)^2} $$
   $$ n = \frac{(2.8)^2 \cdot (0.2175)}{(0.05)^2} $$
   $$ n = \frac{7.84 \cdot 0.2175}{0.0025} $$
   $$ n = \frac{1.7034}{0.0025} = 681.36 $$
   Thus, round up to the nearest whole number: $n = 682$.
   Therefore, a sample size of 682 users is needed.

In practical terms, this means that to detect a statistically significant increase in the user conversion rate from 10% to 15% with 95% confidence and 80% power, Bloomberg would need to sample at least 682 users. This sample size ensures that we are adequately powered to identify the effect of any changes made to the user conversion process while controlling for Type I and Type II errors.

This problem is closely related to concepts in hypothesis testing, particularly the balance between Type I error (false positive) and Type II error (false negative). The methodology also touches on confidence intervals for proportions, which could lead to discussions about how to interpret results when the sample size is limited. Additionally, we could explore multiple testing corrections if multiple conversion strategies are tested simultaneously, such as the Bonferroni correction or the False Discovery Rate (FDR) methods. Understanding these extensions could be important for discussing the robustness of findings in practical applications.

Calculate probability for coin flips

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