Analyze survey with gender imbalance

## Scenario You ran a user survey to measure satisfaction with a new product feature. Each respondent reports: - `gender ∈ {female, male}` - `satisfaction` (e.g., binary satisfied/not satisfied, or a 1–5 Likert score) After data collection, you notice the survey respondents are **not gender-balanced** (e.g., the female/male split in respondents differs from the true split in your user population). ## Questions 1. **What are the key concerns** with using this survey to estimate overall satisfaction for the full user population? 2. **How would you validate** whether the imbalance is problematic (i.e., whether it biases your estimate)? What checks or additional data would you use? 3. **What distributional assumptions** might you make for: - the gender counts in the sample (female vs male), and - the satisfaction outcome within each gender? Explain when those assumptions are reasonable. 4. Compute the probability of observing exactly **30 female respondents out of 70 total respondents**: - (a) if each respondent is sampled independently from a population where the true female proportion is `p`, and - (b) if you sampled *without replacement* from a finite population of size `N` with `F` females. 5. How would **stratified sampling** help here, and how would you analyze the results if you used stratified sampling (including how to combine strata to estimate overall satisfaction)?