Let's analyze each statement one by one and determine whether they are true (T) or false (F).
### Step-by-Step Solution
**Statement (i):** If the number of observations is even, then the median is the mean of the \((n/2)^{th}\) and \((n/2 + 1)^{th}\) terms.
- **Explanation:** When the number of observations (n) is even, the median is calculated by taking the average of the two middle values, which are the \((n/2)^{th}\) and \((n/2 + 1)^{th}\) terms.
- **Conclusion:** This statement is **True (T)**.
---
**Statement (ii):** After four vertical lines for tally marks, if the tally mark occurs for the fifth time, then the fifth line is put vertically with the previous four lines.
- **Explanation:** In tally marking, after four vertical lines, the fifth tally is represented by crossing the previous four lines and drawing a vertical line. Therefore, the statement is incorrect.
- **Conclusion:** This statement is **False (F)**.
---
**Statement (iii):** If the range of the data with minimum value 16 is 87, then the maximum value is 71.
- **Explanation:** The range is calculated as:
\[
\text{Range} = \text{Maximum value} - \text{Minimum value}
\]
Given that the minimum value is 16 and the range is 87:
\[
87 = \text{Maximum value} - 16
\]
Rearranging gives:
\[
\text{Maximum value} = 87 + 16 = 103
\]
Therefore, the maximum value is 103, not 71.
- **Conclusion:** This statement is **False (F)**.
---
**Statement (iv):** Mode of the data 14, 71, 51, 91, 15, 2, 15, 51, 19, 41, 51, 15, 51 is 51.
- **Explanation:** The mode is the value that appears most frequently in the data set. In this case, the number 51 appears four times, which is more than any other number.
- **Conclusion:** This statement is **True (T)**.
---
**Statement (v):** Mean of the first ten natural numbers is 5.5.
- **Explanation:** The first ten natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The mean is calculated as:
\[
\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10}{10}
\]
The sum of the first ten natural numbers can also be calculated using the formula:
\[
\text{Sum} = \frac{n(n + 1)}{2} = \frac{10 \times 11}{2} = 55
\]
Thus, the mean is:
\[
\text{Mean} = \frac{55}{10} = 5.5
\]
- **Conclusion:** This statement is **True (T)**.
---
### Final Answers
- (i) T
- (ii) F
- (iii) F
- (iv) T
- (v) T
So the final answer is **TFFTT**.
