Question

a political pollster selects a random sample of 75 registered voters from allegheny county, pennsylvania. the city of pittsburgh is in allegheny county. the pollster is interested in determining if the distribution of political party preference among registered voters in this county differs between those who live in pittsburgh and those who live in the suburbs. the responses are displayed in the table.

the pollster would like to know if these data provide convincing evidence of a relationship between residential status and political party in the population of all registered voters in allegheny county. the random and 10% conditions are met. is the large counts condition met?
yes, the smallest expected count is 8.64, so all expected counts are at least 5.
yes, the smallest expected count is 9.36, so all expected counts are at least 5.
no, the smallest expected count is 0.20, so the expected counts are not at least 5.
no, the smallest expected count is 1.10, so the expected counts are not all at least 5.

Explanation:

Step1: Calculate row and column totals

Row totals: City: \(20 + 8+11=39\), Suburbs: \(8 + 10+18 = 36\)
Column totals: Democratic: \(20 + 8=28\), Independent: \(8+10 = 18\), Republican: \(11+18=29\)
Total sample size \(n=75\)

Step2: Calculate expected counts formula

The formula for the expected count \(E=\frac{\text{Row total}\times\text{Column total}}{n}\)

For City - Democratic: \(E_{11}=\frac{39\times28}{75}=\frac{1092}{75}=14.56\)
For City - Independent: \(E_{12}=\frac{39\times18}{75}=\frac{702}{75}=9.36\)
For City - Republican: \(E_{13}=\frac{39\times29}{75}=\frac{1131}{75}=15.08\)
For Suburbs - Democratic: \(E_{21}=\frac{36\times28}{75}=\frac{1008}{75}=13.44\)
For Suburbs - Independent: \(E_{22}=\frac{36\times18}{75}=\frac{648}{75}=8.64\)
For Suburbs - Republican: \(E_{23}=\frac{36\times29}{75}=\frac{1044}{75}=13.92\)

Step3: Check the Large Counts condition

The Large Counts condition for a chi - square test of association requires that all expected counts \(E\geq5\). The smallest expected count is \(8.64\)

Answer:

Yes, the smallest expected count is \(8.64\), so all expected counts are at least \(5\)