Question

a politician selects a random sample of 306 voters from her state. for each voter, she records the region in which they live and their political affiliation. she would like to know if there is convincing evidence that the region in which voters live is associated with their political affiliation. let \\( \alpha = 0.05 \\). the observed counts are displayed in the table. are the conditions for inference met? no, the data do not come from a random sample. no, the 10% condition is not met. no, the large counts condition is not met since all expected counts are not greater than 5. all conditions for inference are met.

Explanation:

• Random sample: The problem states that the politician selects a random sample of voters. So the random condition is met.
• 10% condition: We assume that the number of voters in the state is much larger than \(n = 306\) (since \(306<0.1N\) where \(N\) is the total number of voters in the state).
• Large Counts condition:
• First, calculate the row totals: \(56 + 39=95\) (East), \(51+58 = 109\) (Center), \(40+62=102\) (West).
• Column totals: \(56+51 + 40=147\) (Democratic), \(39+58+62 = 159\) (Republican).
• The formula for the expected count \(E_{ij}=\frac{\text{Row total}_i\times\text{Column total}_j}{n}\), where \(n = 306\).
• For East - Democratic: \(E=\frac{95\times147}{306}\approx45.6\)
• For East - Republican: \(E=\frac{95\times159}{306}\approx49.4\)
• For Center - Democratic: \(E=\frac{109\times147}{306}\approx52.4\)
• For Center - Republican: \(E=\frac{109\times159}{306}\approx56.6\)
• For West - Democratic: \(E=\frac{102\times147}{306}\approx49.0\)
• For West - Republican: \(E=\frac{102\times159}{306}\approx53.0\)
• All expected counts \(E>5\).

Answer:

All conditions for inference are met.