This question asks us to find a possible value for 'n' given the sum of 'n' values (\(\sum^n_{i=1} x_i\)) and the sum of the squares of those 'n' values (\(\sum^n_{i=1}x^2_i\)). We are given:

• \(\sum^n_{i=1} x_i = 80\)
• \(\sum^n_{i=1}x^2_i = 400\)

We need to check which of the given options for 'n' (9, 12, 15, 18) is plausible.

Understanding Variance Constraints

A fundamental concept in statistics is that the variance of a dataset must always be non-negative (greater than or equal to zero). Variance measures the spread of the data points around the mean.

The formula for variance (\(\sigma^2\)) can be expressed using the mean (\(\bar{x}\)) and the mean of the squares (\(\overline{x^2}\)) as follows:

\(\sigma^2 = \overline{x^2} - (\bar{x})^2\)

Where:

• \(\bar{x} = \frac{\sum^n_{i=1} x_i}{n}\) (the mean)
• \(\overline{x^2} = \frac{\sum^n_{i=1}x^2_i}{n}\) (the mean of the squares)

Calculating Mean and Mean of Squares

Using the given values, we can calculate the mean and the mean of the squares in terms of 'n':

• Mean (\(\bar{x}\)): \(\bar{x} = \frac{80}{n}\)
• Mean of Squares (\(\overline{x^2}\)): \(\overline{x^2} = \frac{400}{n}\)

Deriving the Inequality for 'n'

Now, we substitute these into the variance formula and apply the constraint that variance must be non-negative (\(\sigma^2 \ge 0\)):

\(\frac{400}{n} - \left(\frac{80}{n}\right)^2 \ge 0\)

Let's simplify this inequality:

\(\frac{400}{n} - \frac{80^2}{n^2} \ge 0\)

\(\frac{400}{n} - \frac{6400}{n^2} \ge 0\)

To solve for 'n', we can multiply the entire inequality by \(n^2\). Since 'n' represents the number of values, it must be a positive integer (\(n > 0\)), so multiplying by \(n^2\) does not change the direction of the inequality:

\(400n - 6400 \ge 0\)

Now, add 6400 to both sides:

\(400n \ge 6400\)

Finally, divide by 400:

\(n \ge \frac{6400}{400}\)

\(n \ge 16\)

This inequality tells us that for the given sums to be possible, the value of 'n' must be 16 or greater.

Checking the Options

We now check the provided options against the condition \(n \ge 16\):

• Option 1: \(n = 9\). Since \(9 < 16\), this is not a possible value.
• Option 2: \(n = 12\). Since \(12 < 16\), this is not a possible value.
• Option 3: \(n = 15\). Since \(15 < 16\), this is not a possible value.
• Option 4: \(n = 18\). Since \(18 \ge 16\), this is a possible value.

Conclusion

Based on the non-negativity constraint of variance, the only value among the options that satisfies \(n \ge 16\) is 18. Therefore, 18 is a possible value for 'n'.