Calculate variance for coin flips

Question

Given the following scenario about user conversion, calculate the the sample size needed.

Codemia · Accepted Answer

To determine the sample size needed for a coin flip experiment, we need to establish our hypotheses and parameters. Let's define our null hypothesis (H0) as the coin being fair, i.e., the probability of heads (p) is 0.5. The alternative hypothesis (H1) is that the coin is biased, with p ≠ 0.5. We also need to set our significance level (alpha), commonly set at 0.05, and the desired power of the test, typically 0.8. The variance of a binomial distribution, which is applicable here since we are dealing with coin flips, is given by the formula:

$$ \sigma^2 = n \cdot p \cdot (1 - p) $$\n
Where $ n $ is the sample size, and $ p $ is the probability of success (getting heads).

1. **Set the parameters**: Let's assume we want to detect a difference of 0.1 in the proportion of heads (p = 0.6 or p = 0.4).
2. **Calculate the required sample size** using the formula for sample size in hypothesis testing for proportions:
   $$ n = \left( \frac{Z_{\alpha/2} + Z_{\beta}}{p_1 - p_0} \right)^2 \cdot p(1 - p) $$\n   Where:  
   - $ Z_{\alpha/2} $ is the Z-score for the significance level (1.96 for alpha = 0.05)  
   - $ Z_{\beta} $ is the Z-score for the desired power (0.84 for power = 0.8)  
   - $ p_0 = 0.5 $ is the null hypothesis proportion (fair coin)  
   - $ p_1 $ is the alternative hypothesis proportion (0.6 for detecting bias).  
3. **Plug in the values and compute n**.

Given the parameters above:  
- $ p_0 = 0.5 $  
- $ p_1 = 0.6 $  
- $ p = 0.5 $  
- $ Z_{\alpha/2} \approx 1.96 $  
- $ Z_{\beta} \approx 0.84 $

Calculate the sample size:  
1. The difference in proportions, $ p_1 - p_0 = 0.6 - 0.5 = 0.1 $.  
2. Plugging into the sample size formula:  
   $$ n = \left( \frac{1.96 + 0.84}{0.1} \right)^2 \cdot 0.5(1 - 0.5) $$  
   $$ n = \left( \frac{2.8}{0.1} \right)^2 \cdot 0.25 $$  
   $$ n = (28)^2 \cdot 0.25 = 784 \cdot 0.25 = 196 $$  
Thus, a sample size of approximately 196 coin flips is required to detect a bias with 80% power at a 5% significance level.

The result indicates that to confidently detect a bias in the coin flips (from a fair 0.5 to a biased 0.6) with a statistical power of 80%, one would need to perform at least 196 flips. This sample size ensures that if the coin is indeed biased, there is an 80% chance of correctly rejecting the null hypothesis, thereby providing actionable insights for decision-making. In practice, this means that if Palantir were to conduct such an experiment, they should prepare for that number of trials to have reliable data.

This problem connects to several advanced statistical concepts:  
1. **Bayesian Methods**: Instead of fixed p-values, one could analyze the posterior distribution of the bias probability given prior beliefs about the coin's fairness.  
2. **Power Analysis**: Understanding how changes in effect size (e.g., bias) or significance level affect required sample size.  
3. **Causal Inference**: Exploring how the design of the experiment could consider confounding factors that might influence the outcome of the coin flips (e.g., environmental conditions).  
4. **Resampling Techniques**: Such as bootstrapping, might provide alternative estimates of sample size or confidence intervals for the observed proportion of heads in the study.

Calculate variance for coin flips

Palantir

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

Palantir 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