The Spearman's rank correlation coefficient `= (9)/(11) `, given ` sum d^(2) = 30 ` . Find the number of observation.

Answer

To find the number of observations (N) given the Spearman's rank correlation coefficient and the sum of squared differences, we can follow these steps:

### Step 1: Write down the formula for Spearman's rank correlation coefficient (P).
The formula is:
\[ P = 1 - \frac{6 \Sigma d^2}{N(N^2 - 1)} \]
where \( \Sigma d^2 \) is the sum of the squared differences of ranks, and \( N \) is the number of observations.

### Step 2: Substitute the known values into the formula.
Given:
- \( P = \frac{9}{11} \)
- \( \Sigma d^2 = 30 \)

Substituting these values into the formula:
\[ \frac{9}{11} = 1 - \frac{6 \times 30}{N(N^2 - 1)} \]

### Step 3: Simplify the equation.
Rearranging the equation gives:
\[ \frac{6 \times 30}{N(N^2 - 1)} = 1 - \frac{9}{11} \]
Calculating \( 1 - \frac{9}{11} \):
\[ 1 - \frac{9}{11} = \frac{11 - 9}{11} = \frac{2}{11} \]
So, we have:
\[ \frac{180}{N(N^2 - 1)} = \frac{2}{11} \]

### Step 4: Cross-multiply to eliminate the fraction.
Cross-multiplying gives:
\[ 180 \times 11 = 2 \times N(N^2 - 1) \]
Calculating \( 180 \times 11 \):
\[ 1980 = 2N(N^2 - 1) \]

### Step 5: Divide both sides by 2.
\[ 990 = N(N^2 - 1) \]

### Step 6: Rewrite the equation.
We can rewrite this as:
\[ N^3 - N - 990 = 0 \]

### Step 7: Find integer solutions for N.
We can try different integer values for \( N \) to find a solution. Testing \( N = 10 \):
\[ 10^3 - 10 - 990 = 1000 - 10 - 990 = 0 \]
Thus, \( N = 10 \) is a solution.

### Conclusion:
The number of observations \( N \) is \( 10 \).

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