Calculate conditional probability for coin flips

Question

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

Codemia · Accepted Answer

We want to calculate the sample size needed to determine conditional probabilities based on coin flips, which we can relate to the revenue per session for a product. Let's define:

- **H0 (Null Hypothesis)**: The probability of revenue per session is the same regardless of the outcome of the coin flip (i.e., p = 0.5).
- **H1 (Alternative Hypothesis)**: The probability of revenue per session differs depending on the outcome of the coin flip (i.e., p ≠ 0.5).

We will use the formula for sample size estimation in hypothesis testing for proportions:

$$ n = \frac{(Z_{\alpha/2} + Z_{\beta})^2 \cdot p(1-p)}{(p_1 - p_0)^2} }{(p_1 - p_0)^2} $$

Where:
- $Z_{\alpha/2}$ is the Z-score for the desired level of significance (e.g., 1.96 for 95% confidence).
- $Z_{\beta}$ is the Z-score for the desired power of the test (e.g., 0.84 for 80% power).
- $p$ is the estimated proportion (0.5 under H0).
- $p_1$ and $p_0$ are the proportions under H1 and H0 respectively.

1. **Define Parameters**: Assume we want to detect a difference in revenue per session where under H0, $p_0 = 0.5$ and we anticipate under H1, $p_1 = 0.6$.
2. **Calculate Z-scores**: For a 95% confidence level, $Z_{\alpha/2} = 1.96$. For an 80% power, $Z_{\beta} = 0.84$.
3. **Substitute into the formula**:
   - Using $p = 0.5$, $p_1 = 0.6$, and $p_0 = 0.5$:
   - $n = \frac{(1.96 + 0.84)^2 \cdot 0.5(1-0.5)}{(0.6 - 0.5)^2} $
4. **Simplify and Calculate**:
   - $n = \frac{(2.8)^2 \cdot 0.25}{(0.1)^2} = \frac{7.84 \cdot 0.25}{0.01} = 196$
   - Thus, a sample size of 196 is required.

Assuming we are looking at a revenue per session and we want to determine if the coin flip affects the outcome:

1. We hypothesize that with a 50% chance (H0) of revenue being influenced, we anticipate an increase to 60% (H1).
2. Plugging in the values:
   - $Z_{\alpha/2} = 1.96, Z_{\beta} = 0.84, p = 0.5, p_1 = 0.6, p_0 = 0.5$
3. Calculation:
   - $n = \frac{(1.96 + 0.84)^2 \cdot 0.5(0.5)}{(0.1)^2} = \frac{7.84 \cdot 0.25}{0.01} = 196$
4. Therefore, we need at least 196 samples to confidently make a decision regarding the revenue per session post-coin flip.

The result indicates that to detect a statistically significant difference in revenue based on the outcome of a coin flip, we need a minimum of 196 observations. Practically, this means that Oracle should ensure they collect data from at least 196 sessions to validate their hypothesis and ensure any conclusions drawn about revenue differences are robust and reliable.

This problem connects to several related concepts:
- **Power Analysis**: Understanding the balance between sample size and the ability to detect a true effect.
- **Bayesian vs Frequentist Approaches**: If we were to use a Bayesian framework, we might incorporate prior beliefs about the revenue outcomes.
- **Causal Inference**: While this problem focuses on correlation through hypothesis testing, understanding the causality behind the coin flips and revenue can deepen insights into customer behavior.
- **Conditional Probability**: This case demonstrates how the outcome of one event (coin flip) can influence the probability of another event (revenue outcome), a key concept in probabilistic modeling.

Calculate conditional probability for coin flips

Oracle

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

Oracle 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