Answer :
All calculations in the provided steps are correct and consistent with the given two-way table. Therefore, no arithmetical error was made.
### Explanation
1. Analyze the problem and data
Let's analyze the two-way table and Joseph's calculations to identify the error. The table represents the social networking habits of classmates regarding Site 1 and Site 2. We need to check each step to ensure the calculations are correct and consistent with the table's totals.
2. Verify Step 1
Step 1: This step calculates the number of students who use Site 1 and Site 2. It subtracts the number of students who do not use Site 1 but use Site 2 (30) from the total number of students who use Site 2 (85). The calculation is $85 - 30 = 55$. This is correct.
3. Verify Step 2
Step 2: This step calculates the number of students who use Site 1 but not Site 2. It subtracts the number of students who use both Site 1 and Site 2 (55, from Step 1) from the total number of students who use Site 1 (80). The calculation is $80 - 55 = 25$. This is correct.
4. Verify Step 3
Step 3: This step calculates the number of students who do not use Site 1. It subtracts the total number of students who use Site 1 (80) from the overall total number of students (140). The calculation is $140 - 80 = 60$. This is correct.
5. Verify Step 4
Step 4: This step calculates the number of students who do not use Site 1 and do not use Site 2. It subtracts the number of students who do not use Site 1 but use Site 2 (30) from the total number of students who do not use Site 1 (60, from Step 3). The calculation is $60 - 30 = 30$. This is correct.
6. Verify Step 5
Step 5: This step calculates the number of students who do not use Site 2. It adds the number of students who use Site 1 but not Site 2 (25, from Step 2) and the number of students who do not use Site 1 and do not use Site 2 (30, from Step 4). The calculation is $25 + 30 = 55$. However, this value should equal to the 'No Site 2' total, which can also be found by subtracting the number of students who use Site 2 (85) from the total number of students (140), i.e. $140 - 85 = 55$. This is correct. However, the problem states that Step 5 should calculate the total number of students who do NOT use Site 2. Looking at the table, this should be the sum of the number of students who use Site 1 but not Site 2 (25) and the number of students who do not use Site 1 and do not use Site 2 (30). So, $25 + 30 = 55$. This matches the total given in the table (140 - 85 = 55). Thus, there is no arithmetical error.
7. Conclusion
Upon reviewing all steps, all calculations are arithmetically correct and consistent with the information provided in the table. Therefore, there is no arithmetical error in any of the steps.
### Examples
Two-way tables are commonly used in surveys and data analysis to understand relationships between different categories. For example, a school might use a two-way table to analyze students' participation in sports and clubs to identify trends and improve student engagement.
### Explanation
1. Analyze the problem and data
Let's analyze the two-way table and Joseph's calculations to identify the error. The table represents the social networking habits of classmates regarding Site 1 and Site 2. We need to check each step to ensure the calculations are correct and consistent with the table's totals.
2. Verify Step 1
Step 1: This step calculates the number of students who use Site 1 and Site 2. It subtracts the number of students who do not use Site 1 but use Site 2 (30) from the total number of students who use Site 2 (85). The calculation is $85 - 30 = 55$. This is correct.
3. Verify Step 2
Step 2: This step calculates the number of students who use Site 1 but not Site 2. It subtracts the number of students who use both Site 1 and Site 2 (55, from Step 1) from the total number of students who use Site 1 (80). The calculation is $80 - 55 = 25$. This is correct.
4. Verify Step 3
Step 3: This step calculates the number of students who do not use Site 1. It subtracts the total number of students who use Site 1 (80) from the overall total number of students (140). The calculation is $140 - 80 = 60$. This is correct.
5. Verify Step 4
Step 4: This step calculates the number of students who do not use Site 1 and do not use Site 2. It subtracts the number of students who do not use Site 1 but use Site 2 (30) from the total number of students who do not use Site 1 (60, from Step 3). The calculation is $60 - 30 = 30$. This is correct.
6. Verify Step 5
Step 5: This step calculates the number of students who do not use Site 2. It adds the number of students who use Site 1 but not Site 2 (25, from Step 2) and the number of students who do not use Site 1 and do not use Site 2 (30, from Step 4). The calculation is $25 + 30 = 55$. However, this value should equal to the 'No Site 2' total, which can also be found by subtracting the number of students who use Site 2 (85) from the total number of students (140), i.e. $140 - 85 = 55$. This is correct. However, the problem states that Step 5 should calculate the total number of students who do NOT use Site 2. Looking at the table, this should be the sum of the number of students who use Site 1 but not Site 2 (25) and the number of students who do not use Site 1 and do not use Site 2 (30). So, $25 + 30 = 55$. This matches the total given in the table (140 - 85 = 55). Thus, there is no arithmetical error.
7. Conclusion
Upon reviewing all steps, all calculations are arithmetically correct and consistent with the information provided in the table. Therefore, there is no arithmetical error in any of the steps.
### Examples
Two-way tables are commonly used in surveys and data analysis to understand relationships between different categories. For example, a school might use a two-way table to analyze students' participation in sports and clubs to identify trends and improve student engagement.
