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adverse reactions of a drug used to help patients continue their abstinence from the use of alcohol have been studied in clinical trials, and the accompanying table summarizes results for the digestive system effects among patients from different treatment groups. Use a 0.01 significance level to test the claim that experiencing an adverse reaction in the digestive system is independent of the treatment group. Does the drug treatment appear to have an effect on the digestive system? Determine the test statistic and round to three decimal places based on the data pictured above.

To test whether experiencing an adverse reaction in the digestive system is independent of the treatment group, we'll perform a **Chi-Square Test for Independence**.

### Step 1: Set up hypotheses
- **Null Hypothesis \( H_0 \)**: The adverse reaction is independent of the treatment group (no association).
- **Alternative Hypothesis \( H_a \)**: The adverse reaction is dependent on the treatment group (association exists).

### Step 2: Data extraction

From the table provided in the image, the data is as follows:

|                       | Placebo | Drug 1380 mg | Drug 2070 mg | Total |
|-----------------------|---------|--------------|--------------|-------|
| **Adverse Effect**     | 338     | 90           | 8            | 436   |
| **No Adverse Effect**  | 1434    | 769          | 77           | 2280  |
| **Total**              | 1772    | 859          | 85           | 2716  |

### Step 3: Expected frequencies
We calculate the expected frequency for each cell using the formula:

\[
E = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}
\]

For example:
- The expected frequency for "Adverse Effect, Placebo" = \( \frac{436 \times 1772}{2716} \approx 284.51 \)

We will calculate the expected frequencies for all cells.

### Step 4: Chi-Square Test Statistic
The Chi-Square test statistic is calculated as:

\[
\chi^2 = \sum \frac{(O - E)^2}{E}
\]

where \( O \) is the observed frequency, and \( E \) is the expected frequency.

Let me calculate these values for you.The test statistic for the Chi-Square test is approximately **34.603**.

### Conclusion:
With this test statistic, we can compare it to a critical value from the Chi-Square distribution or calculate the corresponding p-value to determine if the drug treatment has a statistically significant effect on the digestive system.

Do you need further details or clarification on how to continue with this analysis?

### Follow-up questions:
1. How do we calculate the degrees of freedom for the Chi-Square test?
2. How is the critical value for a Chi-Square test determined?
3. What is the p-value, and how do we interpret it in this context?
4. Could other statistical tests be applied to this data?
5. How does sample size affect the power of a Chi-Square test?

**Tip:** Always round the test statistic to three decimal places for consistency and precision when presenting results.

Adverse reactions of a drug used to help patients continue their abstinence from alcohol have been studied in clinical trials, and the table summarizes results for digestive system effects among patients from different treatment groups. Use a 0.01 significance level to test the claim that experiencing an adverse reaction in the digestive system is independent of the treatment group. Does the drug treatment appear to have an effect on the digestive system? Determine the test statistic and round to three decimal places based on the data.

Learn how to perform a Chi-Square test for independence to determine if there is a statistically significant association between drug treatments and adverse effects on the digestive system. Includes detailed data analysis from clinical trials comparing placebo and different dosages of a drug used to help alcohol abstinence. Suitable for college-level statistics students.

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