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To solve this problem, we need to complete the table and conduct a chi-square test to determine if the observed results significantly differ from the expected results.

### Step-by-Step Solution:

1. **Calculate Differences (o - e):**
   - For AA: \( 60 - 70 = -10 \)
   - For A: \( 20 - 20 = 0 \)
   - For B: \( 20 - 10 = 10 \)

2. **Square of Differences \((o - e)^2\):**
   - For AA: \((-10)^2 = 100\)
   - For A: \(0^2 = 0\)
   - For B: \(10^2 = 100\)

3. **\((o - e)^2 / e\) Calculation:**
   - For AA: \(100 / 70 = 1.43\)
   - For A: \(0 / 20 = 0\)
   - For B: \(100 / 10 = 10\)

4. **Sum \((\Sigma)\):**
   - Add up the values in the \((o - e)^2 / e\) column:
     \[
     1.43 + 0 + 10 = 11.43
     \]

### Degrees of Freedom:
- Since there are 3 categories, the degrees of freedom \( A = 3 - 1 = 2 \).

### Chi-Square Critical Value:
- For \( p = 0.05 \) and \( \text{df} = 2 \), the critical value from the chi-square table is \( 5.99 \).

### Decision:
- Compare the calculated chi-square value (11.43) with the critical value (5.99).
- Since \( 11.43 > 5.99 \), we **reject the null hypothesis**.

### Final Answer:
\[
\text{11.4,reject}
\]

Would you like more details on any step, or do you have further questions?

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Here are five follow-up questions related to this topic:

1. What would happen if the observed values were closer to the expected values?
2. How would you interpret a situation where the chi-square value is less than the critical value?
3. How do degrees of freedom affect the chi-square test?
4. What are other common uses of the chi-square test besides goodness of fit?
5. How would the conclusion change if we used a different significance level, such as \( p = 0.01 \)?

**Tip:** When interpreting chi-square results, always compare your calculated chi-square value with the critical value for your chosen significance level and degrees of freedom.

The farm normally produced three grades of eggs in the proportions shown below in the #expected column. As part of normal quality control measures, a sample of 100 eggs was collected, which had the observed amounts shown. Complete the table, and determine whether the sample was significantly different from the expected results (accept or reject null hypothesis).

Learn how to use the chi-square test to determine if observed egg quality distributions differ from expected values. This example problem involves calculating differences, squaring, dividing by expected counts, and comparing to a critical chi-square value for hypothesis testing. Suitable for students in Grades 10-12.

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