Calculate the​ chi-squared test statistic.

χ^2equals=enter your response here 


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.

Learn how to calculate the chi-squared (\( \chi^2 \)) test statistic step-by-step with an example. This problem involves using observed and expected frequencies to compute the chi-squared value, crucial in statistical analysis and hypothesis testing.

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