Calculate variance for click-through rates

Question

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

Codemia · Accepted Answer

The analytical question here is to determine the sample size needed to accurately assess the variance in click-through rates (CTR) for Slack's revenue per session. To address this, we need historical data on click-through rates, ideally segmented by user demographics, session types, and any other relevant categories. We would also require the average CTR and the standard deviation of CTR from previous sessions to inform our sample size calculation.

For this problem, we'll use the formula for calculating the sample size needed for estimating a proportion, which is suitable for CTR. The formula is:

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

Where:
- $ n $ = required sample size
- $ Z $ = Z-score (for a 95% confidence level, Z is approximately 1.96)
- $ p $ = estimated proportion (previous CTR)
- $ E $ = margin of error (the precision we want, e.g., 0.01 for 1%)

Assuming Slack's current average CTR is 0.15 (15%) and we want a margin of error of 0.01 (1%). Plugging in the values:

$$ n = \frac{(1.96)^2 \cdot 0.15(1 - 0.15)}{(0.01)^2} $$
$$ n = \frac{3.8416 \cdot 0.15 \cdot 0.85}{0.0001} $$
$$ n = \frac{0.4894}{0.0001} \approx 4894 $$

Thus, we'd need approximately 4,894 sessions to achieve our desired confidence level and margin of error.

The result indicates that to reliably estimate the CTR with a 1% margin of error at a 95% confidence level, Slack should aim for nearly 4,900 sessions. This means that any analysis conducted with fewer sessions may lead to inaccurate conclusions about user engagement and revenue implications. Additionally, it sets the groundwork for understanding how changes in the product may influence CTR.

Common pitfalls specific to this analysis include:
- Failing to account for variability in different segments, which could lead to underestimating the required sample size. If CTR varies significantly across demographics, a single average may not suffice.
- Not considering the ecological fallacy; assuming that observed CTR trends in aggregated data will hold true at the individual level.
- Ignoring potential novelty effects or interference, where the introduction of a new feature may temporarily boost or dampen CTR, skewing results if not properly randomized.

Calculate variance for click-through rates

Slack

What the Interviewer Expects

Key Topics to Cover

How to Approach This

Possible Follow-up Questions

Sharpen Your Skills on Codemia

Sample Answer

Problem Setup

Methodology

Submit Your Answer

Slack Product Manager Interview Guide

Related Questions

Explain Bayesian vs frequentist with an example

Explain p-value with an example

Interpret no significant difference from an experiment

Design an A/B test for a recommendation algorithm

Calculate conditional probability for user retention