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A professor is trying to determine if her students guessed on a certain multiple choice question. She expects that if the students guessed, the distribution of answers would be uniform for that question. She compares the observed distribution of answers with the uniform distribution. Find the test statistic, χ20, for the chi-square goodness-of-fit test. If necessary, round the final answer to three decimal places.

χ20=∑k(O−E)2E

Answer Choice

A

B

C

D

Expected

10

10

10

10

Observed

11

9

8

12

Provide your answer below:

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.

Learn how to calculate the chi-square goodness-of-fit test statistic using observed and expected frequencies. This problem involves comparing observed data against expected uniform distribution, suitable for statistical analysis.

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