To solve for the test statistic $\chi^2_0$ using the chi-square goodness-of-fit test formula:

$\chi^2_0 = \sum \frac{(O - E)^2}{E}$

where $O$ is the observed frequency, and $E$ is the expected frequency.

Given:

• Expected frequencies ($E$): 10 for each option (A, B, C, D)
• Observed frequencies ($O$):
  • A: 11
  • B: 9
  • C: 8
  • D: 12

Step-by-Step Calculation:

1. For Option A: $\frac{(O - E)^2}{E} = \frac{(11 - 10)^2}{10} = \frac{1^2}{10} = \frac{1}{10} = 0.1$

2. For Option B: $\frac{(9 - 10)^2}{10} = \frac{(-1)^2}{10} = \frac{1}{10} = 0.1$

3. For Option C: $\frac{(8 - 10)^2}{10} = \frac{(-2)^2}{10} = \frac{4}{10} = 0.4$

4. For Option D: $\frac{(12 - 10)^2}{10} = \frac{2^2}{10} = \frac{4}{10} = 0.4$

Sum of the above results:

$\chi^2_0 = 0.1 + 0.1 + 0.4 + 0.4 = 1.0$

Final Answer:

The test statistic $\chi^2_0$ is 1.0.

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. What is the critical value for the chi-square distribution at a certain significance level?
2. How do you determine the degrees of freedom in a chi-square goodness-of-fit test?
3. What assumptions must be met for the chi-square goodness-of-fit test to be valid?
4. How would the test statistic change if one observed frequency was much larger than expected?
5. What are some real-world applications of the chi-square goodness-of-fit test?
6. How do you interpret the p-value obtained from the chi-square test?
7. What happens if the expected frequency in a category is very low?
8. Can the chi-square goodness-of-fit test be used for data that is not categorical?

Tip: The chi-square test is sensitive to the sample size; larger samples can detect smaller deviations from the expected distribution.