Certainly! Let's go through the problem step-by-step to find the probable error.
### Step 1: Understand the Given Values
You are given:
- [tex]\( r = \frac{2}{\sqrt{10}} \)[/tex]
- [tex]\( n = 36 \)[/tex]
### Step 2: Simplify the Value of [tex]\( r \)[/tex]
First, let's simplify the expression for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{2}{\sqrt{10}} \][/tex]
We can rationalize the denominator:
[tex]\[ r = \frac{2}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{2\sqrt{10}}{10} = \frac{\sqrt{10}}{5} \][/tex]
Calculating this value:
[tex]\[ r = \frac{2}{\sqrt{10}} \approx 0.6324555320336759 \][/tex]
### Step 3: Probable Error Formula
The formula to calculate the probable error (P.E.) is:
[tex]\[ \text{P.E.} = 0.6745 \cdot \frac{1 - r^2}{\sqrt{n}} \][/tex]
### Step 4: Calculate [tex]\( r^2 \)[/tex]
[tex]\[ r^2 = \left(\frac{2}{\sqrt{10}}\right)^2 = \left(0.6324555320336759\right)^2 = 0.4 \][/tex]
### Step 5: Subtract [tex]\( r^2 \)[/tex] from 1
[tex]\[ 1 - r^2 = 1 - 0.4 = 0.6 \][/tex]
### Step 6: Calculate the Square Root of [tex]\( n \)[/tex]
[tex]\[ \sqrt{n} = \sqrt{36} = 6 \][/tex]
### Step 7: Divide [tex]\( 1 - r^2 \)[/tex] by [tex]\( \sqrt{n} \)[/tex]
[tex]\[ \frac{1 - r^2}{\sqrt{n}} = \frac{0.6}{6} = 0.1 \][/tex]
### Step 8: Multiply by [tex]\( 0.6745 \)[/tex] to Find the P.E.
[tex]\[ \text{P.E.} = 0.6745 \times 0.1 = 0.06745 \][/tex]
So, the probable error is [tex]\( 0.06745 \)[/tex].
### Summary
Given [tex]\( r = \frac{2}{\sqrt{10}} \approx 0.6324555320336759 \)[/tex] and [tex]\( n = 36 \)[/tex], the probable error is [tex]\( 0.06745 \)[/tex].
