To calculate the chi-squared ($\chi^2$) test statistic, we'll use the formula:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

where:

• $O_i$ is the observed frequency.
• $E_i$ is the expected frequency under the null hypothesis.

Step 1: Calculate the expected frequencies

The expected frequency for each cell in a contingency table is calculated using the formula:

$E_i = \frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}$

Let's calculate the expected frequencies for each cell:

1. For Placebo & Yes: $E_{Placebo, Yes} = \frac{681 \times 46}{1353} \approx 23.16$
2. For Placebo & No: $E_{Placebo, No} = \frac{681 \times 1307}{1353} \approx 657.84$
3. For Aspirin & Yes: $E_{Aspirin, Yes} = \frac{672 \times 46}{1353} \approx 22.84$
4. For Aspirin & No: $E_{Aspirin, No} = \frac{672 \times 1307}{1353} \approx 649.16$

Step 2: Calculate the chi-squared statistic

Now we use the observed frequencies and the expected frequencies to calculate $\chi^2$:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

$\chi^2 = \frac{(28 - 23.16)^2}{23.16} + \frac{(653 - 657.84)^2}{657.84} + \frac{(18 - 22.84)^2}{22.84} + \frac{(654 - 649.16)^2}{649.16}$

Let's calculate each term:

1. $\frac{(28 - 23.16)^2}{23.16} \approx \frac{23.2064}{23.16} \approx 1.002$
2. $\frac{(653 - 657.84)^2}{657.84} \approx \frac{23.3856}{657.84} \approx 0.036$
3. $\frac{(18 - 22.84)^2}{22.84} \approx \frac{23.3856}{22.84} \approx 1.024$
4. $\frac{(654 - 649.16)^2}{649.16} \approx \frac{23.2064}{649.16} \approx 0.036$

Adding them together:

$\chi^2 \approx 1.002 + 0.036 + 1.024 + 0.036 = 2.098$

Final Answer:

$\chi^2 \approx 2.098$

Would you like to know how to interpret this $\chi^2$ value, or have any questions?

Here are 8 related questions that might interest you:

1. How do you interpret the $\chi^2$ value in terms of statistical significance?
2. What are the degrees of freedom for this $\chi^2$ test?
3. How can you find the p-value corresponding to the calculated $\chi^2$ statistic?
4. What is the critical $\chi^2$ value for a given significance level (e.g., 0.05)?
5. What does a small $\chi^2$ value indicate about the relationship between the treatment and response?
6. How would the results change if the observed and expected frequencies were closer?
7. How can you perform a $\chi^2$ test in statistical software like R or Python?
8. What other tests could be used to assess the relationship between treatment and response?

Tip: Always check if the expected frequencies are sufficiently large (typically at least 5) when applying the $\chi^2$ test.