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)?