Calculate expected value for user retention

Question

Given the following scenario about revenue per session, calculate the the p-value.

Codemia · Accepted Answer

To calculate the expected value for user retention based on revenue per session, we need to define the parameters of our study. We will form two hypotheses:
- **Null Hypothesis (H0)**: There is no significant difference in user retention before and after a change in the product.
- **Alternative Hypothesis (H1)**: There is a significant difference in user retention after the change.

Let $ R $ be the revenue per session, and we collect data on user retention rates before and after the change. Our goal is to analyze this data to calculate the p-value, which tells us the likelihood of observing our results (or more extreme) given that H0 is true.

1. **Data Collection**: Gather user retention rates and revenue per session data for both pre-change (Group A) and post-change (Group B).
2. **Calculate Means**: Compute the mean retention rates for both groups: 
   - $ ar{R_A} = 	ext{mean revenue per session for Group A} $
   - $ ar{R_B} = 	ext{mean revenue per session for Group B} $
3. **Conduct a t-test**: Use a t-test to compare the means of the two groups to see if there is a statistically significant difference. 
4. **Calculate p-value**: The t-test will yield a t-statistic, which can be used to find the p-value corresponding to the observed difference in means.

Assume we have the following data:
- Group A (pre-change): Retention rate = 70%, Revenue per session = $5
- Group B (post-change): Retention rate = 80%, Revenue per session = $6

1. Means:
   - $ ar{R_A} = 5 $
   - $ ar{R_B} = 6 $

2. Perform a t-test:
   - Calculate the t-statistic using the formula:
   
   $$ t = \frac{\bar{R_B} - \bar{R_A}}{s_p \sqrt{\frac{1}{n_A} + \frac{1}{n_B}}} $$
   
   where $ s_p $ is the pooled standard deviation.

3. Assume we calculate $ t $ and find a p-value of 0.04. This means there is a 4% probability of observing such a difference in retention rates (or more extreme) if H0 is true.

A p-value of 0.04 indicates that there is moderate evidence against the null hypothesis (H0). In practical terms, it suggests that the change made to the product may have positively impacted user retention. Since this p-value is less than the common alpha level of 0.05, we would reject H0 and conclude that the change likely improved user retention rates.

This analysis connects to various statistical concepts, such as:
- **Power Analysis**: To determine how large of a sample size is needed to detect a meaningful effect size, we could conduct a power analysis prior to the study.
- **Multiple Testing Correction**: If we were testing multiple changes at once, we would need to apply corrections like Bonferroni or False Discovery Rate (FDR) to maintain our overall Type I error rate.
- **Regression Analysis**: We could extend this analysis by using regression models to account for other variables that might affect user retention, such as session length or user demographics.

Calculate expected value for user retention

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