To find the standard deviation of the observations \(15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25\) given that the standard deviation of the observations \(-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\) is \(\sqrt{10}\), we can follow these steps:
### Step 1: Understand the relationship between the two sets of observations
The first set of observations is \(-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\). The second set of observations is \(15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25\).
### Step 2: Identify the transformation
Notice that each element in the second set can be obtained by adding \(20\) to each element in the first set:
- \(-5 + 20 = 15\)
- \(-4 + 20 = 16\)
- \(-3 + 20 = 17\)
- \(-2 + 20 = 18\)
- \(-1 + 20 = 19\)
- \(0 + 20 = 20\)
- \(1 + 20 = 21\)
- \(2 + 20 = 22\)
- \(3 + 20 = 23\)
- \(4 + 20 = 24\)
- \(5 + 20 = 25\)
### Step 3: Apply the property of standard deviation
When a constant is added to each observation in a dataset, the standard deviation remains unchanged. Therefore, the standard deviation of the second set of observations will be the same as that of the first set.
### Step 4: Conclude the standard deviation
Since the standard deviation of the first set is given as \(\sqrt{10}\), the standard deviation of the second set will also be:
\[
\text{Standard Deviation of } (15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25) = \sqrt{10}
\]
### Final Answer
The standard deviation of the observations \(15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25\) is \(\sqrt{10}\).
---
