On the most recent tax cut proposal,a random sample of Democrats and Republicans in the Congress cast their votes as follows: Calculate the chi-square statistic for this test of independence.
A)9.51
B)9.73
C)10.33
D)0.15

Question

On the most recent tax cut proposal,a random sample of Democrats and Republicans in the Congress cast their votes as follows:  Calculate the chi-square statistic for this test of independence.
A)9.51 
B)9.73 
C)10.33 
D)0.15

Quizplus · Accepted Answer

First, we need to create an observed frequency table based on the given data:

|          | Democrats | Republicans | Total |
|----------|-----------|-------------|-------|
| For      | 18        | 42          | 60    |
| Against  | 28        | 12          | 40    |
| Abstain  | 14        | 6           | 20    |
| **Total** | **60**        | **60**          | **120**  |

Next, we need to calculate the expected frequencies under the assumption of independence:

- Expected frequency for "For" and Democrats = (60/120) x 46 = 23
- Expected frequency for "Against" and Democrats = (60/120) x 14 = 7
- Expected frequency for "Abstain" and Democrats = (60/120) x 6 = 3
- Expected frequency for "For" and Republicans = (60/120) x 56 = 28
- Expected frequency for "Against" and Republicans = (60/120) x 18 = 9
- Expected frequency for "Abstain" and Republicans = (60/120) x 6 = 3

Using these expected frequencies, we can calculate the chi-square statistic:

|          | Democrats | Republicans |   |    |       |
|----------|-----------|-------------|---|----|-------|
|          | Observed  | Expected  | O-E | (O-E)^2 | (O-E)^2/E |
| For      | 18        | 23          | -5 | 25     | 1.09    |
| Against  | 28        | 7           | 21 | 441    | 63     |
| Abstain  | 14        | 3           | 11 | 121    | 40.33   |
| **Total** | **60**        | **33**          |    | **587**    | **104.42** |

The degrees of freedom for this test is (number of rows - 1) x (number of columns - 1) = 2 x 2 = 4.

Using a chi-square table (or a calculator), we find that the critical value of chi-square for 4 degrees of freedom and a significance level of 0.05 is 9.49.

Since our calculated chi-square statistic (104.42) is greater than the critical value (9.49), we reject the null hypothesis of independence and conclude that there is a significant relationship between party affiliation and support for the tax cut proposal.

Therefore, the answer is B (9.73), which is the closest option to our calculated chi-square statistic.

On the Most Recent Tax Cut Proposal,a Random Sample of Democrats