Calculate expected value for user retention

Question

Given the following scenario about revenue per session, calculate the the sample size needed.

Codemia · Accepted Answer

To calculate the expected value for user retention, we need to formulate the problem with specific parameters. Let:
- $ R $ = Revenue per session (in dollars)
- $ p $ = Probability of user retention per session
- $ N $ = Sample size needed to achieve a specific confidence level for our estimates.

The expected revenue from a retained user can be expressed as:
$$ E(R) = N \cdot R \cdot p $$\
Our goal is to determine the sample size $ N $ such that we can reliably estimate the expected revenue from retained users within a specified margin of error at a given confidence level (e.g., 95%).

1. **Define Parameters**: Let's assume the following:
   - Average revenue per session $ R = 10 $ dollars.
   - Estimated probability of user retention $ p = 0.6 $.
   - Desired margin of error $ E = 1 $ dollar.
   - Confidence level = 95%, corresponding to a z-score of approximately 1.96.

2. **Calculate Sample Size**: Use the formula for sample size in estimating a proportion:
$$ N = \left( \frac{z^2 \cdot p(1-p)}{E^2} \right) $$\
   Substituting the values:
$$ N = \left( \frac{(1.96)^2 \cdot (0.6)(0.4)}{(1)^2} \right) $$\
   $$ N = \left( \frac{3.8416 \cdot 0.24}{1} \right) \approx 0.922 \approx 923 \text{ (rounding up)} $$\
   Therefore, we need a sample size of approximately 923 sessions.

Assuming we want to estimate the revenue based on a sample of users:
1. Given our revenue per session $ R = 10 $ dollars and retention probability $ p = 0.6 $, we expect:
   - Expected number of retained sessions from 923 users = $ 923 \cdot 0.6 = 553.8 $ sessions.
2. The expected revenue from retained users:
   - Total expected revenue = $ 553.8 \cdot 10 = 5538 $ dollars.
This means that if we sample 923 users, we expect to retain approximately 554 users, generating around 5538 dollars in revenue.

The calculated sample size of 923 sessions allows us to estimate user retention and associated revenue with a margin of error of 1 dollar at a 95% confidence level. This means if we were to conduct this experiment multiple times, 95% of the time our estimate of expected revenue from retained users would fall within 1 dollar of the true mean. Practically, this is crucial for Robinhood as it aids in decision-making regarding marketing strategies and user engagement initiatives.

This problem connects to several related concepts:
- **Multiple Testing Correction**: If we were testing multiple user retention strategies simultaneously, we would need to apply corrections like Bonferroni to maintain acceptable error rates.
- **Conditional Probability and Bayes Theorem**: Understanding how different user segments may retain differently could involve Bayesian approaches, updating probabilities based on new data.
- **Causal Inference Basics**: Establishing whether changes in user engagement lead to higher retention could require A/B testing frameworks to infer causality rather than mere correlation.
- **Confidence Intervals and Significance Levels**: Beyond point estimates, understanding the confidence intervals around our revenue estimates allows for more nuanced business decisions.

Calculate expected value for user retention

Robinhood

What the Interviewer Expects

Key Topics to Cover

How to Approach This

Possible Follow-up Questions

Sharpen Your Skills on Codemia

Sample Answer

Problem Formulation

Solution Approach

Submit Your Answer

Robinhood Data Scientist Interview Guide

Related Questions

Interpret mixed results across segments from an experiment

Explain instrumental variables with an example

Calculate conditional probability for coin flips

Explain difference-in-differences with an example

Explain p-value with an example