Here's a clear, educational, and well-structured solution to the problem:

1. Key Concepts and Formulas

   To solve this problem, we need to understand the definitions and formulas for mean and variance, especially when combining two sets of observations.

   • Mean ($\bar{x}$): The average of $N$ observations $x_1, x_2, \dots, x_N$ is given by $\bar{x} = \frac{\sum_{i=1}^{N} x_i}{N}$. From this, we can find the sum of observations: $\sum x_i = N \bar{x}$.
   • Variance ($\sigma^2$): A measure of data dispersion, calculated as $\sigma^2 = \frac{\sum_{i=1}^{N} x_i^2}{N} - (\bar{x})^2$. From this, we can find the sum of squares of observations: $\sum x_i^2 = N(\sigma^2 + (\bar{x})^2)$.
   • Combined Statistics: When two sets of observations are combined (Set I: $N_1, \bar{x}_1, \sigma_1^2$; Set II: $N_2, \bar{x}_2, \sigma_2^2$), the combined statistics are:
     • Total Number of Observations: $N_{comb} = N_1 + N_2$
     • Combined Mean: $\bar{x}_{comb} = \frac{N_1 \bar{x}_1 + N_2 \bar{x}_2}{N_1 + N_2}$
     • Combined Variance: $\sigma^2_{comb} = \frac{\sum X^2_{comb}}{N_{comb}} - (\bar{x}_{comb})^2$, where $\sum X^2_{comb} = N_1(\sigma_1^2 + \bar{x}_1^2) + N_2(\sigma_2^2 + \bar{x}_2^2)$.

2. Step-by-Step Solution

   We are given the statistics for two sets of observations and the variance of their combined set. Our goal is to find the size ($n$) of the second set.

   Step 1: Calculate the sum of observations and sum of squares for Set I.

   • What we are doing: We are extracting the fundamental sums ($\sum x_i$ and $\sum x_i^2$) for the first set. These are essential building blocks for calculating the combined statistics.
   • Given for Set I:
     • Size ($N_1$) = 10
     • Mean ($\bar{x}_1$) = 2
     • Variance ($\sigma_1^2$) = 2
   • Calculation:
     1. Sum of observations ($\sum x_1$): $\sum x_1 = N_1 \bar{x}_1 = 10 \times 2 = 20$
     2. Sum of squares of observations ($\sum x_1^2$): $\sum x_1^2 = N_1(\sigma_1^2 + \bar{x}_1^2) = 10(2 + (2)^2) = 10(2 + 4) = 10 \times 6 = 60$
   • Reasoning: The sum of observations allows us to find the total sum for the combined set. The sum of squares is directly used in the formula for combined variance.

   Step 2: Calculate the sum of observations and sum of squares for Set II (in terms of $n$).

   • What we are doing: Similarly, we are finding the fundamental sums for the second set, which depend on the unknown size $n$.
   • Given for Set II:
     • Size ($N_2$) = $n$
     • Mean ($\bar{x}_2$) = 3
     • Variance ($\sigma_2^2$) = 1
   • Calculation:
     1. Sum of observations ($\sum x_2$): $\sum x_2 = N_2 \bar{x}_2 = n \times 3 = 3n$
     2. Sum of squares of observations ($\sum x_2^2$): $\sum x_2^2 = N_2(\sigma_2^2 + \bar{x}_2^2) = n(1 + (3)^2) = n(1 + 9) = n \times 10 = 10n$
   • Reasoning: These expressions in terms of $n$ will be combined with the values from Set I to form equations for the combined set, which we can then solve for $n$.

   Step 3: Calculate the combined statistics.

   • What we are doing: We are now combining the information from both sets to find the total number of observations, the total sum of observations, the total sum of squares, and the combined mean.
   • Calculation:
     1. Total number of observations ($N_{comb}$): $N_{comb} = N_1 + N_2 = 10 + n$
     2. Total sum of observations ($\sum X_{comb}$): $\sum X_{comb} = \sum x_1 + \sum x_2 = 20 + 3n$
     3. Total sum of squares of observations ($\sum X^2_{comb}$): $\sum X^2_{comb} = \sum x_1^2 + \sum x_2^2 = 60 + 10n$
     4. Combined Mean ($\bar{x}_{comb}$): $\bar{x}_{comb} = \frac{\sum X_{comb}}{N_{comb}} = \frac{20 + 3n}{10 + n}$
   • Reasoning: These combined values are directly substituted into the combined variance formula to form an equation in terms of $n$. The combined mean is a necessary intermediate step for the combined variance calculation.

   Step 4: Apply the combined variance formula and solve for $n$.

   • What we are doing: We use the given combined variance and the expressions derived in Step 3 to form an algebraic equation and solve for $n$.
   • Given combined variance: $\sigma^2_{comb} = \frac{17}{9}$
   • Formula: $\sigma^2_{comb} = \frac{\sum X^2_{comb}}{N_{comb}} - (\bar{x}_{comb})^2$
   • Substitution and Calculation: $\frac{17}{9} = \frac{60 + 10n}{10 + n} - \left(\frac{20 + 3n}{10 + n}\right)^2$ To simplify, find a common denominator $(10+n)^2$: $\frac{17}{9} = \frac{(60 + 10n)(10 + n)}{(10 + n)^2} - \frac{(20 + 3n)^2}{(10 + n)^2}$ $\frac{17}{9} = \frac{(60 + 10n)(10 + n) - (20 + 3n)^2}{(10 + n)^2}$ Expand the numerator:
     • $(60 + 10n)(10 + n) = 600 + 60n + 100n + 10n^2 = 10n^2 + 160n + 600$
     • $(20 + 3n)^2 = 400 + 120n + 9n^2$ Substitute back into the equation: $\frac{17}{9} = \frac{(10n^2 + 160n + 600) - (9n^2 + 120n + 400)}{(10 + n)^2}$ Carefully distribute the negative sign: $\frac{17}{9} = \frac{10n^2 + 160n + 600 - 9n^2 - 120n - 400}{(10 + n)^2}$ Combine like terms in the numerator: $\frac{17}{9} = \frac{n^2 + 40n + 200}{(10 + n)^2}$ Cross-multiply: $17(10 + n)^2 = 9(n^2 + 40n + 200)$ Expand $(10+n)^2 = 100 + 20n + n^2$: $17(100 + 20n + n^2) = 9n^2 + 360n + 1800$ Distribute constants: $1700 + 340n + 17n^2 = 9n^2 + 360n + 1800$ Rearrange into a quadratic equation $an^2 + bn + c = 0$: $17n^2 - 9n^2 + 340n - 360n + 1700 - 1800 = 0$ $8n^2 - 20n - 100 = 0$ Divide by 4 to simplify: $2n^2 - 5n - 25 = 0$ Factor the quadratic equation: We look for two numbers that multiply to $2 \times (-25) = -50$ and add to $-5$. These are $-10$ and $5$. $2n^2 - 10n + 5n - 25 = 0$ $2n(n - 5) + 5(n - 5) = 0$ $(2n + 5)(n - 5) = 0$ This gives two possible solutions for $n$: $2n + 5 = 0 \implies n = -\frac{5}{2}$ $n - 5 = 0 \implies n = 5$
   • Reasoning: Since $n$ represents the number of observations, it must be a positive integer. Therefore, $n=5$ is the only valid solution.

3. Common Mistakes & Tips

   • Algebraic Errors: Be extremely careful when expanding squared terms like $(20+3n)^2$ and distributing negative signs, e.g., $-(9n^2 + 120n + 400)$ becomes $-9n^2 - 120n - 400$. This is a frequent source of errors.
   • Formula Application: Ensure you use the correct variance formula and its derived form for $\sum x^2$. A common mistake is using $\sum x^2 = N \sigma^2$ instead of $\sum x^2 = N(\sigma^2 + \bar{x}^2)$.
   • Physical Constraints: Always check if your final answer makes sense in the context of the problem. $n$ must be a positive integer, as it represents the count of observations.

4. Summary

   This problem involved calculating the mean and variance for two individual sets of observations, then combining them to find the overall variance. By systematically using the definitions of mean and variance to find the sums and sums of squares for each set, we were able to set up an equation for the combined variance. Solving the resulting quadratic equation yielded $n=5$ as the physically meaningful number of observations.

The final answer is $\boxed{5}$.