Pollsters want to test if an association exists between a person’s profession and their political party. A random sample of 295295 voters is polled, resulting in the data in the following table. Based on these results, is there enough evidence at the 0.010.01 level of significance to say that an association exists? Observed Sample of VotersDemocratRepublicanTotalDoctor

45

22

67

Lawyer

22

19

41

Teacher

36

38

74

Farmer

44

24

68

Laborer

28

17

45

Total

175

120

295 Calculate the expected value for the number of doctors voting for the Democratic party. Round your answer to six decimal places.

We are tasked with calculating the expected value for the number of doctors voting for the Democratic party using a contingency table for a chi-square test of independence.

### Given dataWe are tasked with calculating the expected value for the number of doctors voting for the Democratic party using a contingency table for a chi-square test of independence.

### Given 
| Profession | Democrat | Republican | Total |
|------------|----------|------------|-------|
| Doctor     | 45       | 22         | 67    |
| Lawyer     | 22       | 19         | 41    |
| Teacher    | 36       | 38         | 74    |
| Farmer     | 44       | 24         | 68    |
| Laborer    | 28       | 17         | 45    |
| **Total**  | **175**  | **120**    | **295**|

We are asked to compute the **expected value** for the number of doctors voting for the Democratic party.

### Formula for Expected Value:
The formula to calculate the expected value \( E_{ij} \) for a cell in a contingency table is:

\[
E_{ij} = \frac{(\text{Row Total of } i) \times (\text{Column Total of } j)}{\text{Grand Total}}
\]

Here:
- Row total for doctors: 67
- Column total for Democrats: 175
- Grand total: 295

Substituting these values into the formula:

\[
E_{Doctor, Democrat} = \frac{67 \times 175}{295}
\]

Let's calculate that.The expected value for the number of doctors voting for the Democratic party is approximately **39.745763**.

Would you like further details or have any questions about the calculations?

Here are some related questions:
1. How do we interpret the chi-square test of independence?
2. What is the next step in conducting the full chi-square test for this data?
3. How do you compute the chi-square statistic for the entire table?
4. What is the significance level and how is it applied in hypothesis testing?
5. Can this test determine causation or just association?

**Tip:** Always ensure the assumptions of a chi-square test are met, such as having expected frequencies greater than 5 in each cell.

Pollsters want to test if an association exists between a person’s profession and their political party. A random sample of 295 voters is polled, resulting in the data in the following table. Based on these results, is there enough evidence at the 0.01 level of significance to say that an association exists? Calculate the expected value for the number of doctors voting for the Democratic party. Round your answer to six decimal places.

This problem involves calculating the expected value for the number of doctors voting for the Democratic party in a sample of 295 voters. Learn how to apply the chi-square test of independence, use contingency tables, and calculate expected values. Suitable for Grades 11-12 or students studying statistics.

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