We have a contingency table for the students based on whether they took the course and the number of incidents they were involved in. Here, we need to determine the expected frequencies for the three missing cells, assuming that the two variables (course status and lab performance) are independent.

Step 1: Calculate Marginal Totals

• Total for “Took the techniques course”: $34 + 28 + 11 = 73$
• Total for “Didn’t take the techniques course”: $69 + 35 + 23 = 127$
• Total for “Involved in no incident”: $34 + 69 = 103$
• Total for “Involved in exactly one incident”: $28 + 35 = 63$
• Total for “Involved in 2+ incidents”: $11 + 23 = 34$

Step 2: Calculate Expected Frequencies

We use the formula for the expected frequency in a contingency table, given by:

$\text{Expected Frequency} = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}$

Here, the Grand Total is $200$.

1. For the cell “Took the techniques course” and “Involved in no incident”: $\text{Expected Frequency} = \frac{(73) \times (103)}{200} = \frac{7519}{200} = 37.595$

2. For the cell “Took the techniques course” and “Involved in exactly one incident”: $\text{Expected Frequency} = \frac{(73) \times (63)}{200} = \frac{4599}{200} = 22.995$

3. For the cell “Took the techniques course” and “Involved in 2+ incidents”: $\text{Expected Frequency} = \frac{(73) \times (34)}{200} = \frac{2482}{200} = 12.41$

Step 3: Round the Answers

• Involved in no incident (Took the course): $37.60$
• Involved in exactly one incident (Took the course): $22.99$
• Involved in 2+ incidents (Took the course): $12.41$

Final Answer:

1. Involved in no incident: 37.60
2. Involved in exactly one incident: 22.99
3. Involved in 2+ incidents: 12.41

Would you like additional details or have any questions? Here are some relative questions to further explore:

1. How can we use the chi-square test to verify the independence of two categorical variables?
2. What are marginal totals in a contingency table and how do they help in calculations?
3. Why is the assumption of independence crucial in calculating expected frequencies?
4. How do we interpret the expected values compared to observed values in a contingency table?
5. What is the formula for calculating the chi-square statistic based on observed and expected frequencies?

Tip: When calculating expected frequencies, always double-check the row, column, and grand totals to avoid rounding errors.