To answer this question, we need to fill in the missing value in the table under "Bus to school" for high school students in such a way that there is no evidence of an association between the variables "bus to school" and "drive to school." In statistical terms, this would imply that the data is consistent with no relationship between these variables, which can be tested using the concept of expected frequencies in a contingency table.

Steps to calculate the expected value:

1. Total for middle school students:

   • Bus to school: 256
   • Drive to school: 64
   • Total = 256 + 64 = 320

2. Total for high school students:

   • Bus to school: Unknown (this is what we need to find)
   • Drive to school: 78
   • Total = Unknown + 78

3. Column totals:

   • Total for students taking the bus to school: 256 + Unknown
   • Total for students driving to school: 64 + 78 = 142

4. Overall total:

   • The total number of students can be found by adding the known totals: $256 + 64 + \text{Unknown} + 78 = 398 + \text{Unknown}$

To find the value that suggests no association, we would typically expect the ratio of middle to high school students to be consistent across the two methods of transportation. This can be done by calculating the expected frequencies using the formula:

$\text{Expected frequency for bus to school (high school)} = \frac{(\text{Total for high school}) \times (\text{Total for bus to school})}{\text{Overall total}}$

Let me calculate the expected value.The expected value for the number of high school students taking the bus to school, assuming no association between the variables, is approximately 206.

Thus, the value closest to this would be option B: 78, which suggests no strong evidence of an association between the variables.

Would you like to dive deeper into the details of this calculation?

Here are 5 related questions:

1. How can contingency tables help in understanding relationships between variables?
2. What is the concept of independence in statistical terms?
3. How is the chi-square test used to measure associations?
4. Why is the expected frequency calculation important in data analysis?
5. What assumptions are made when testing for no association between variables?

Tip: Always ensure that the totals in the contingency table match after inserting the expected value to avoid any inconsistencies.