Calculate probability for coin flips

Question

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

Codemia · Accepted Answer

To determine the sample size needed for our experiment at Robinhood, we first need to define the analytical question clearly: *What is the effect of different revenue per session metrics on user engagement?*

The data required for this analysis includes:
- Historical revenue data per session for users.
- The expected effect size (the minimum difference in revenue per session that we want to detect).
- The standard deviation of revenue per session based on past data.
- Desired power of the test (commonly set at 0.80).
- Significance level (alpha, commonly set at 0.05).

These elements will allow us to calculate the necessary sample size to achieve statistically significant results.

For this analysis, we will employ a power analysis using the two-sample t-test to calculate the required sample size. This method is appropriate because we are comparing the means of two independent groups (e.g., users exposed to different revenue structures).

The formula for calculating sample size (n) is:

$$ n = 2 \left( \frac{(Z_{\alpha/2} + Z_{\beta})^2 \cdot (\sigma^2_1 + \sigma^2_2)}{(\mu_1 - \mu_2)^2} \right) \right] $$

Where:
- $Z_{\alpha/2}$ is the z-score corresponding to the desired significance level.
- $Z_{\beta}$ is the z-score corresponding to the desired power.
- $\sigma^2_1$ and $\sigma^2_2$ are the variances of the two groups.
- $\mu_1$ and $\mu_2$ are the expected means of the two groups.

Let's assume:
- We expect a minimum detectable effect size of $10 in revenue per session (\mu_1 - \mu_2 = 10)$.
- The historical standard deviation of revenue per session is $30 (\sigma = 30)$.
- We want a significance level of 0.05 and a power of 0.80.

From the z-tables, we find:
- $Z_{0.025} \approx 1.96$ (for a two-tailed test at alpha = 0.05)
- $Z_{0.20} \approx 0.84$ (for power = 0.80)

Substituting these values into the sample size formula:

$$ n = 2 \left( \frac{(1.96 + 0.84)^2 \cdot (30^2 + 30^2)}{10^2} \right) \right] $$
$$ n = 2 \left( \frac{(2.8)^2 \cdot (900 + 900)}{100} \right) \right] $$
$$ n = 2 \left( \frac{7.84 \cdot 1800}{100} \right) \right] $$
$$ n = 2 \left( 141.12 \right) \approx 282.24 \implies 283 \text{ participants per group.} $$

Thus, we would need approximately 283 participants in each group for a total of 566 participants.

The calculated sample size of 566 participants allows us to detect a difference of at least $10 in revenue per session with a power of 80% and a significance level of 5%. This means we can be reasonably confident that if we observe a difference at this level, it is likely not due to chance.

In the context of Robinhood, this analysis helps ensure that any change in revenue structure is meaningful and can lead to informed decisions about user engagement strategies.

Common pitfalls in this type of analysis include:
- Underestimating the required sample size, leading to insufficient power and inconclusive results.
- Failing to account for potential confounding variables that may affect revenue per session, such as user demographics or market conditions.
- Ignoring the assumptions of the t-test, such as normality and homogeneity of variance, which could invalidate results.
- Not conducting a sensitivity analysis to see how changes in expected effect size or standard deviation impact sample size requirements.

Calculate probability for coin flips

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