Imagine we have a binary indicator variable that is equal to 1 if an individual is employed and zero otherwise. We can retrieve the fraction who are employed by:

Please mark all that apply!

Question

Imagine we have a binary indicator variable that is equal to 1 if an individual is employed and zero otherwise. We can retrieve the fraction who are employed by:

Please mark all that apply!

BlackTom AI · Accepted Answer

For a binary indicator variable equal to 1 for employed and 0 otherwise, the proportion employed equals the mean of that variable.

1) Relevant knowledge points and concepts
- A binary (indicator) variable xi takes values 1 (success/employed) or 0 (failure/unemployed).
- The sample mean of a variable is (1/N) Σ xi.
- For a binary variable, Σ xi counts the number of 1's, so the mean = (number of 1's)/N, which is the fraction employed.
- The ratio (number of 1's)/(number of 0's) is the odds, not the fraction.

2) Step-by-step reasoning
- Let N be total observations and k be the number of individuals with xi = 1.
- Sum of the observations: Σ xi = 1+1+... (k times) + 0+0+... (N−k times) = k.
- The sample average: x̄ = (1/N) Σ xi = k/N.
- Hence the fraction employed = k/N.

3) Calculations with intermediate steps (symbolic and numeric)
- Symbolic: x̄ = (1/N) Σ xi = k/N.
- Numeric example: N = 100, k = 30 employed.
  - Counting 1's and dividing by total: 30/100 = 0.30.
  - Taking the average: (30·1 + 70·0)/100 = 30/100 = 0.30.

4) Why the correct answers are correct
- "Counting the number of individuals with 1's and dividing by the total number of observations" yields k/N directly.
- "Taking the average of the variable" computes (1/N) Σ xi which equals k/N for a 0/1 variable. Both procedures produce the same fraction employed.

5) Why the other options are incorrect
- "Count the number of total individuals and then divide by the number with 1s" computes N/k (e.g., 100/30 ≈ 3.33), which is not a proportion and is >1 when k<N.
- "Counting the number of individuals with a 1 and divide by the number of individuals with a zero" computes k/(N−k), the odds, not the fraction employed (e.g., 30/70 ≈ 0.4286).

6) Conclusion
Both "counting the number of individuals with 1's and dividing by the total number of observations" and "taking the average of the variable" are correct. This matches the provided answer.

Imagine we have a binary indicator variable that is equal to 1 if an individual is employed and zero otherwise. We can retrieve the fraction who are employed by: Please mark all that apply!Multiple choice

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