To ensure statistical independence, we need to use the concept of expected frequencies. Two variables are independent when the joint probability of two events equals the product of their individual probabilities.

The table you've given represents the following structure:

	Good	Bad	Total
New	100	200
Old	175	?
Total

Steps for solving:

1. Calculate the totals for each row and column:

   • Total for New customers: $100 + 200 = 300$
   • Total for Old customers: $175 + ?$
   • Total for Good ratings: $100 + 175$
   • Total for Bad ratings: $200 + ?$

2. Sum the totals for the entire table:

   • Total number of observations = $100 + 200 + 175 + ?$

3. Apply the formula for independence: For statistical independence, the expected frequency of the intersection (cell) should be the product of the marginal probabilities:

   $P(\text{New and Good}) = P(\text{New}) \times P(\text{Good})$

   In terms of frequencies, this translates to:

   $E(\text{New and Good}) = \frac{(\text{Total New}) \times (\text{Total Good})}{\text{Total population}}$

4. Solve for the unknown: With the equation above, we can substitute the known values to solve for the missing frequency in the table.

Let me compute this for you! Known values good_new = 100 bad_new = 200 good_old = 175

Total for New and Old categories

total_new = good_new + bad_new total_old = good_old + 175 # To be calculated

Total for Good and Bad categories

total_good = good_new + good_old total_bad = bad_new + 200 # We are looking for this value

Total population

total_population = total_new + total_old

Applying independence formula to find missing value:

expected_bad_old = (total_old * total_bad) / total_population

expected_bad_oldThe value that must have been in the empty cell for statistical independence to hold is approximately 215.38.

This means the number of "Bad" ratings from Old customers would be about 215.38 to ensure the data is statistically independent.

Would you like further explanation or more details? Here are some related questions that might interest you:

1. How do we calculate the expected frequencies for a contingency table?
2. What does it mean for two variables to be independent in a statistical context?
3. How can the Chi-square test be used to check for independence in categorical data?
4. What assumptions do we make when using the concept of statistical independence?
5. How would the calculation change if we had missing data in more than one cell?

Tip: When performing independence tests with contingency tables, remember that the Chi-square statistic helps determine if the observed frequencies differ significantly from the expected frequencies under the assumption of independence.