To determine if a sample is representative, we need to compare the proportion of its characteristics to the proportion of those characteristics in the overall population.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To determine if a sample is representative, we need to compare the proportion of its characteristics to the proportion of those characteristics in the overall population.
Step 1: Analyze the Hypothetical Population.
The population consists of circles (•) and triangles (▲).
Number of circles: 6
Number of triangles: 6
Total population size: 12
Proportion of circles: $\frac{6}{12} = 0.5$ (50%)
Proportion of triangles: $\frac{6}{12} = 0.5$ (50%)
A representative sample should ideally reflect this 1:1 ratio or 50% for each shape.
Step 2: Analyze Sample 1.
Number of circles: 3
Number of triangles: 3
Total sample size: 6
Proportion of circles: $\frac{3}{6} = 0.5$ (50%)
Proportion of triangles: $\frac{3}{6} = 0.5$ (50%)
The proportions in Sample 1 match the population proportions.
Representative? $\boxed{\text{Y}}$
Step 3: Analyze Sample 2.
Number of circles: 4
Number of triangles: 2
Total sample size: 6
Proportion of circles: $\frac{4}{6} = \frac{2}{3} \approx 0.67$ (67%)
Proportion of triangles: $\frac{2}{6} = \frac{1}{3} \approx 0.33$ (33%)
The proportions in Sample 2 (67% circles, 33% triangles) do not match the population proportions (50% circles, 50% triangles).
Representative? $\boxed{\text{N}}$
Step 4: Analyze Sample 3.
Number of circles: 3
Number of triangles: 3
Total sample size: 6
Proportion of circles: $\frac{3}{6} = 0.5$ (50%)
Proportion of triangles: $\frac{3}{6} = 0.5$ (50%)
The proportions in Sample 3 match the population proportions.
Representative? $\boxed{\text{Y}}$
