Calculate variance for user retention

Question

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

Codemia · Accepted Answer

To calculate the sample size needed for evaluating user retention at Slack, we need to establish a clear mathematical framework. We denote the proportion of users retained as $ p $ and the desired confidence level as $ 1 - \alpha $ (e.g., 95% confidence level corresponds to $ \alpha = 0.05 $). We also need the margin of error, $ E $, which is the acceptable difference between the sample proportion and the population proportion. The formula for the sample size $ n $ required to estimate a proportion is given by:

$$ n = \frac{Z^2 \cdot p(1-p)}{E^2} $$

where $ Z $ is the z-score corresponding to the desired confidence level.

To proceed with the calculation, we first need to determine the following components:
1. **Z-score**: For a 95% confidence level, the Z-score is approximately 1.96.
2. **Proportion of retained users (p)**: Assume from previous data that 70% of users are retained, so $ p = 0.7 $.
3. **Margin of Error (E)**: Let's assume we want a margin of error of 5%, so $ E = 0.05 $.

Substituting these values into the formula will allow us to calculate the necessary sample size.

Substituting the values into the sample size formula:

$$ n = \frac{(1.96)^2 \cdot 0.7(1-0.7)}{(0.05)^2} $$

Calculating this step-by-step:
1. Calculate $ p(1-p) $:  $ 0.7 \cdot 0.3 = 0.21 $.
2. Calculate $ Z^2 $:  $ (1.96)^2 \approx 3.8416 $.
3. Now plug these into the formula:
   $$ n = \frac{3.8416 \cdot 0.21}{0.0025} \approx \frac{0.806736}{0.0025} \approx 322.6944 $$

Thus, rounding up, we find that a sample size of approximately 323 users is needed to achieve the desired precision in estimating user retention.

The calculated sample size of 323 users means that if Slack were to conduct a study on user retention, they should aim to collect data from at least this number of users to ensure that the results are statistically significant with a 95% confidence level and a 5% margin of error. This level of rigor helps Slack make informed decisions about user engagement strategies, understanding that the findings can be generalized to the larger user base.

This problem connects to several related concepts, such as:
- **Power analysis**: Understanding the likelihood of detecting an effect if there is one, and how sample size affects power.
- **Causal inference**: Considering how various factors might influence user retention beyond just the retention rate.
- **Impact of confounders**: Identifying potential confounding variables (e.g., changes in product features or external market factors) and discussing strategies for controlling them, such as stratified sampling or randomization.

Calculate variance for user retention

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