1. Key Concepts and Formulas

This problem involves combining two sets of data and calculating their overall variance. The essential statistical concepts and formulas required are:

• Mean ($\bar{X}$): A measure of central tendency, representing the average of a set of numbers.
• Variance ($\sigma^2$): A measure of the spread or dispersion of a set of numbers around its mean. It is the average of the squared differences from the mean.
• Combined Mean Formula: When two groups of data are combined, the mean of the combined set is a weighted average of their individual means. If Group 1 has $n_1$ observations and mean $\bar{X}_1$, and Group 2 has $n_2$ observations and mean $\bar{X}_2$, then the combined mean $\bar{X}_{combined}$ is: $\bar{X}_{combined} = \frac{n_1\bar{X}_1 + n_2\bar{X}_2}{n_1 + n_2}$
• Combined Variance Formula: The variance of the combined set not only depends on the individual variances but also on how far each group's mean deviates from the overall combined mean. The formula for combined variance $\sigma^2_{combined}$ is: $\sigma^2_{combined} = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}$ where:
  • $n_1, n_2$ are the number of observations in Group 1 and Group 2, respectively.
  • $\sigma_1^2, \sigma_2^2$ are the variances of Group 1 and Group 2, respectively.
  • $d_1 = \bar{X}_1 - \bar{X}_{combined}$ is the deviation of Group 1's mean from the combined mean.
  • $d_2 = \bar{X}_2 - \bar{X}_{combined}$ is the deviation of Group 2's mean from the combined mean. The terms $d_1^2$ and $d_2^2$ account for the additional spread introduced by the difference in means between the groups.

2. Step-by-Step Solution

Step 1: Identify the Given Information

First, let's clearly list all the known values provided in the problem statement for both sets of numbers and the combined set.

• For the first set of numbers (Group 1):

  • Number of observations ($n_1$) = 15
  • Mean ($\bar{X}_1$) = 12
  • Variance ($\sigma_1^2$) = 14

• For the second set of numbers (Group 2):

  • Number of observations ($n_2$) = 15
  • Mean ($\bar{X}_2$) = 14
  • Variance ($\sigma_2^2$) = $\sigma^2$ (This is the unknown we need to find)

• For the combined set of 30 numbers:

  • Total number of observations ($n_{combined}$) = $n_1 + n_2 = 15 + 15 = 30$
  • Combined variance ($\sigma^2_{combined}$) = 13

Step 2: Calculate the Combined Mean ($\bar{X}_{combined}$)

Why this step? The formula for combined variance requires the combined mean, $\bar{X}_{combined}$, to calculate the squared deviations ($d_1^2$ and $d_2^2$). Therefore, this is a crucial preliminary calculation.

Using the formula for combined mean: $\bar{X}_{combined} = \frac{n_1\bar{X}_1 + n_2\bar{X}_2}{n_1 + n_2}$ Substitute the known values: $\bar{X}_{combined} = \frac{(15 \times 12) + (15 \times 14)}{15 + 15}$ $\bar{X}_{combined} = \frac{180 + 210}{30}$ $\bar{X}_{combined} = \frac{390}{30}$ $\bar{X}_{combined} = 13$ So, the combined mean of all 30 numbers is 13.

Step 3: Calculate the Squared Deviations ($d_1^2$ and $d_2^2$)

Why this step? These terms quantify how much each group's mean deviates from the overall combined mean. They are essential components of the combined variance formula, capturing the spread due to differences in group averages.

For the first set: $d_1 = \bar{X}_1 - \bar{X}_{combined} = 12 - 13 = -1$ $d_1^2 = (-1)^2 = 1$

For the second set: $d_2 = \bar{X}_2 - \bar{X}_{combined} = 14 - 13 = 1$ $d_2^2 = (1)^2 = 1$

Step 4: Apply the Combined Variance Formula and Solve for $\sigma^2$

Why this step? We now have all the necessary components to use the combined variance formula. This formula includes our unknown variable $\sigma^2$. By substituting all known values, we can form an equation and solve for $\sigma^2$.

Recall the combined variance formula: $\sigma^2_{combined} = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}$ Now, substitute all the values we have identified and calculated:

• $\sigma^2_{combined} = 13$
• $n_1 = 15$
• $\sigma_1^2 = 14$
• $d_1^2 = 1$
• $n_2 = 15$
• $\sigma_2^2 = \sigma^2$ (our unknown)
• $d_2^2 = 1$
• $n_1 + n_2 = 30$

Plugging these into the formula: $13 = \frac{15(14 + 1) + 15(\sigma^2 + 1)}{30}$

Now, let's simplify and solve for $\sigma^2$: Multiply both sides by 30: $13 \times 30 = 15(15) + 15(\sigma^2 + 1)$ $390 = 225 + 15\sigma^2 + 15$ Combine constant terms on the right side: $390 = 240 + 15\sigma^2$ Subtract 240 from both sides: $390 - 240 = 15\sigma^2$ $150 = 15\sigma^2$ Divide both sides by 15: $\sigma^2 = \frac{150}{15}$ $\sigma^2 = 10$

Thus, the variance of the second set of numbers, $\sigma^2$, is 10.

3. Common Mistakes & Tips

• Don't forget the Combined Mean: Many students directly substitute individual means into the combined variance formula without first calculating the overall combined mean. This leads to incorrect $d_i$ values and thus an incorrect combined variance.
• Square the Deviations ($d_i$): Ensure that the deviations of the group means from the combined mean are squared ($d_1^2, d_2^2$) before adding them to the individual variances. Failing to square these terms is a common error.
• Variance vs. Standard Deviation: Remember that the problem provides and asks for variance ($\sigma^2$), not standard deviation ($\sigma$). Do not take a square root unnecessarily.
• Arithmetic Accuracy: These problems involve several arithmetic steps. Double-check all calculations to avoid small errors that can propagate and lead to a wrong final answer.
• Non-negativity of Variance: Variance is always a non-negative value. If your calculation yields a negative variance, it's a definite sign of an error.

4. Summary

To find the unknown variance of the second set, we first calculated the combined mean of all 30 numbers. This combined mean was then used to determine the squared deviations of each group's mean from the overall combined mean. Finally, by substituting all known values into the combined variance formula and solving the resulting equation, we successfully isolated and found the value of $\sigma^2$. The variance of the second set of numbers is 10.

The final answer is $\boxed{\text{10}}$, which corresponds to option (C).