Sets and Standard Deviation

The question provides information about two sets of data, \(x_1, x_2, ..., x_{100}\) and \(y_1, y_2, ..., y_{150}\).

• The first set, x, contains 100 elements. The number of elements is \(n_x = 100\).
• The second set, y, contains 150 elements. The number of elements is \(n_y = 150\).
• Both sets have a mean of zero. This means \(\bar{x} = 0\) and \(\bar{y} = 0\).
• Both sets have the same standard deviation. Let's denote this standard deviation by \(\sigma\).

We are asked to find the ratio of the sum of squares of elements in set x (\(\sum_{i=1}^{100} x_i^2\)) to the sum of squares of elements in set y (\(\sum_{j=1}^{150} y_j^2\)).

Standard Deviation Formula with Zero Mean

The standard deviation (\(\sigma\)) of a set of \(n\) values \(z_1, z_2, ..., z_n\) with a mean \(\bar{z}\) is typically calculated using the formula:

\(\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^n (z_i - \bar{z})^2}\)

Squaring both sides gives the variance (\(\sigma^2\)):

\(\sigma^2 = \frac{1}{n} \sum_{i=1}^n (z_i - \bar{z})^2\)

When the mean (\(\bar{z}\)) is zero, the term \((z_i - \bar{z})\) simplifies to \(z_i - 0 = z_i\). So, for a set with zero mean, the variance formula becomes:

\(\sigma^2 = \frac{1}{n} \sum_{i=1}^n z_i^2\)

From this simplified formula, we can rearrange to find the sum of squares:

\(\sum_{i=1}^n z_i^2 = n \sigma^2\)

Calculating Sum of Squares for Each Set

Using the derived formula \(\sum z_i^2 = n \sigma^2\):

For the first set (x), with \(n_x = 100\) elements and standard deviation \(\sigma\):

\(\sum_{i=1}^{100} x_i^2 = n_x \sigma^2 = 100 \sigma^2\)

For the second set (y), with \(n_y = 150\) elements and the same standard deviation \(\sigma\):

\(\sum_{j=1}^{150} y_j^2 = n_y \sigma^2 = 150 \sigma^2\)

Determining the Ratio

We need to find the ratio of \(\sum_{i=1}^{100} x_i^2\) to \(\sum_{j=1}^{150} y_j^2\). This is the ratio \(\frac{\sum_{i=1}^{100} x_i^2}{\sum_{j=1}^{150} y_j^2}\).

Substitute the calculated sums of squares:

Ratio = \(\frac{100 \sigma^2}{150 \sigma^2}\)

Since both sets have the same standard deviation, \(\sigma\), the \(\sigma^2\) term cancels out (assuming \(\sigma \neq 0\), otherwise the sum of squares would be 0 for both sets, leading to an undefined ratio unless interpreted as 0:0 which is not an option):

Ratio = \(\frac{100}{150}\)

Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 50:

Ratio = \(\frac{100 \div 50}{150 \div 50} = \frac{2}{3}\)

The ratio is \(\frac{2}{3}\), which can be written as 2 ∶ 3.

Comparing with Options

Let's compare our calculated ratio 2 ∶ 3 with the given options:

• Option 1: 1 ∶ 1 (Ratio is 1/1)
• Option 2: √2 ∶ √3 (Ratio is \(\frac{\sqrt{2}}{\sqrt{3}}\))
• Option 3: 2 ∶ 3 (Ratio is 2/3)
• Option 4: 4 ∶ 9 (Ratio is 4/9)

Our calculated ratio 2 ∶ 3 matches Option 3.