Key Concepts and Formulas

1. Effect of Linear Transformation on Mean and Variance: If each element $x_i$ in a dataset is transformed to $x_i' = ax_i + b$ (where $a$ and $b$ are constants), then the new mean ($\mu'$) and variance ($\sigma'^2$) are related to the original mean ($\mu$) and variance ($\sigma^2$) as follows:

   • New Mean: $\mu' = a\mu + b$
   • New Variance: $\sigma'^2 = a^2\sigma^2$. Note that the additive constant $b$ does not affect the variance.

2. Combining Mean and Variance of Two Datasets: For two datasets, say $A'$ and $B'$, with number of elements $n_{A'}$ and $n_{B'}$, means $\mu_{A'}$ and $\mu_{B'}$, and variances $\sigma_{A'}^2$ and $\sigma_{B'}^2$ respectively, when combined to form a new set $C$:

   • Combined Mean ($\mu_C$): The mean of the combined set is the weighted average of the individual means: $\mu_C = \frac{n_{A'}\mu_{A'} + n_{B'}\mu_{B'}}{n_{A'} + n_{B'}}$
   • Combined Variance ($\sigma_C^2$): This requires using the formula for variance $\sigma^2 = \frac{\sum x_i^2}{n} - \mu^2$, which can be rearranged to find the sum of squares: $\sum x_i^2 = n(\sigma^2 + \mu^2)$. The combined variance is then calculated as: $\sigma_C^2 = \frac{\sum_{x \in C} x^2}{n_C} - \mu_C^2$ where $\sum_{x \in C} x^2 = \sum_{x \in A'} x^2 + \sum_{x \in B'} x^2$.

Step-by-Step Solution

We are given information about sets A and B, and we need to find the mean and variance of a new set C, which is formed by transforming elements of A and B and then combining them.

Given Information:

• Set A: $n_A = 5$, $\mu_A = 5$, $\sigma_A^2 = 12$.
• Set B: $n_B = 5$, $\mu_B = 8$, $\sigma_B^2 = 20$.

Step 1: Calculate the mean and variance of the transformed set A (let's call it A'). The elements of A are transformed by subtracting 3 from each. This is a linear transformation $x' = x - 3$, where $a=1$ and $b=-3$.

• Calculate the mean of A' ($\mu_{A'}$): We use the formula $\mu' = a\mu + b$. $\mu_{A'} = 1 \cdot \mu_A - 3 = 1 \cdot 5 - 3 = 2$ Reasoning: Subtracting a constant from every element shifts the entire dataset, so the mean also shifts by that same constant amount.

• Calculate the variance of A' ($\sigma_{A'}^2$): We use the formula $\sigma'^2 = a^2\sigma^2$. $\sigma_{A'}^2 = (1)^2 \cdot \sigma_A^2 = 1 \cdot 12 = 12$ Reasoning: Subtracting a constant from every element does not change the spread or dispersion of the data points relative to each other or to the mean. Thus, the variance remains unchanged.

• The number of elements in A' remains $n_{A'} = 5$.

Step 2: Calculate the mean and variance of the transformed set B (let's call it B'). The elements of B are transformed by adding 2 to each. This is a linear transformation $y' = y + 2$, where $a=1$ and $b=2$.

• Calculate the mean of B' ($\mu_{B'}$): We use the formula $\mu' = a\mu + b$. $\mu_{B'} = 1 \cdot \mu_B + 2 = 1 \cdot 8 + 2 = 10$ Reasoning: Adding a constant to every element shifts the entire dataset, so the mean also shifts by that same constant amount.

• Calculate the variance of B' ($\sigma_{B'}^2$): We use the formula $\sigma'^2 = a^2\sigma^2$. $\sigma_{B'}^2 = (1)^2 \cdot \sigma_B^2 = 1 \cdot 20 = 20$ Reasoning: Adding a constant to every element does not change the spread or dispersion of the data points. Thus, the variance remains unchanged.

• The number of elements in B' remains $n_{B'} = 5$.

Summary of Transformed Sets:

• Set A': $n_{A'} = 5$, $\mu_{A'} = 2$, $\sigma_{A'}^2 = 12$.
• Set B': $n_{B'} = 5$, $\mu_{B'} = 10$, $\sigma_{B'}^2 = 20$.

Step 3: Calculate the mean of the combined set C ($\mu_C$). Set C is formed by combining all elements from A' and B'. The total number of elements in C is $n_C = n_{A'} + n_{B'} = 5 + 5 = 10$.

• Calculate $\mu_C$: We use the formula for the combined mean: $\mu_C = \frac{n_{A'}\mu_{A'} + n_{B'}\mu_{B'}}{n_{A'} + n_{B'}}$. $\mu_C = \frac{(5)(2) + (5)(10)}{5 + 5} = \frac{10 + 50}{10} = \frac{60}{10} = 6$ Reasoning: The combined mean is a weighted average of the individual means, where the weights are the number of elements in each sub-set.

Step 4: Calculate the variance of the combined set C ($\sigma_C^2$). To find the combined variance, we first need to calculate the sum of squares of the elements for A' and B', and then for C. Recall the formula: $\sum x^2 = n(\sigma^2 + \mu^2)$.

• Calculate the sum of squares for A' ($\sum_{x \in A'} x^2$): $\sum_{x \in A'} x^2 = n_{A'}(\sigma_{A'}^2 + \mu_{A'}^2) = 5(12 + 2^2) = 5(12 + 4) = 5(16) = 80$ Reasoning: This formula allows us to find the sum of the squares of the actual data points from the given mean and variance.

• Calculate the sum of squares for B' ($\sum_{y \in B'} y^2$): $\sum_{y \in B'} y^2 = n_{B'}(\sigma_{B'}^2 + \mu_{B'}^2) = 5(20 + 10^2) = 5(20 + 100) = 5(120) = 600$ Reasoning: Similarly, we find the sum of squares for the elements in B'.

• Calculate the total sum of squares for C ($\sum_{z \in C} z^2$): $\sum_{z \in C} z^2 = \sum_{x \in A'} x^2 + \sum_{y \in B'} y^2 = 80 + 600 = 680$ Reasoning: Since set C is the union of A' and B', the sum of squares of its elements is simply the sum of the individual sums of squares.

• Calculate the variance of C ($\sigma_C^2$): We use the formula $\sigma_C^2 = \frac{\sum_{z \in C} z^2}{n_C} - \mu_C^2$. $\sigma_C^2 = \frac{680}{10} - 6^2 = 68 - 36 = 32$ Reasoning: This is the standard formula for variance, using the combined sum of squares and the combined mean.

Step 5: Calculate the sum of the mean and variance of C. We need to find $\mu_C + \sigma_C^2$. $\mu_C + \sigma_C^2 = 6 + 32 = 38$

Common Mistakes & Tips

• Incorrectly combining variances: A common error is to simply average the variances of the individual sets (e.g., $\frac{\sigma_{A'}^2 + \sigma_{B'}^2}{2}$). This is incorrect because variance depends on the deviations from the combined mean, not the individual means. Always use the sum of squares method for combined variance.
• Forgetting $a^2$ in variance transformation: Remember that when scaling data ($x' = ax + b$), the variance is scaled by $a^2$, not just $a$. The additive constant $b$ has no effect on variance.
• Careful with signs: Pay attention to whether a constant is being added or subtracted from the elements when applying linear transformations to the mean.

Summary

We first calculated the mean and variance of the transformed sets A' and B' using the properties of linear transformations on statistical measures. Then, we combined these transformed sets to form set C. For the combined set C, we calculated its mean using the weighted average formula. Finally, to find the variance of C, we calculated the sum of squares for A' and B', summed them to get the total sum of squares for C, and then applied the variance formula. The sum of the mean and variance of C was found to be 38.

The final answer is $\boxed{38}$, which corresponds to option (C).