1. Key Concepts and Formulas

• Mean ($\mu$) for a Frequency Distribution: The mean is the weighted average of the observations, where each observation is weighted by its frequency. $\mu = \frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i} = \frac{\sum f_i x_i}{N}$ where $x_i$ are the observations, $f_i$ are their corresponding frequencies, and $N = \sum f_i$ is the total frequency.
• Variance ($\sigma^2$) for a Frequency Distribution: Variance measures the spread or dispersion of the data. A computationally efficient formula for variance is: $\sigma^2 = \frac{\sum_{i=1}^{n} f_i x_i^2}{N} - \mu^2$
• Solving Systems of Linear Equations: This problem requires solving two linear equations with two unknowns to find the frequencies.

2. Step-by-Step Solution

Step 1: Formulate Equation 1 using the Initial Mean

• What we're doing: We are using the given initial mean ($\mu_1 = 6$) and the structure of the initial frequency distribution to establish the first algebraic relationship between the unknown frequencies, $\alpha$ and $\beta$.
• Why: The mean formula directly links the sum of (frequency × observation) and the total frequency to the given mean, providing a necessary equation to solve for the unknowns.

The initial frequency distribution is: $x: \{x_1=2, x_2=6, x_3=8, x_4=9\}$ $f: \{f_1=4, f_2=4, f_3=\alpha, f_4=\beta\}$

1. Calculate the total frequency ($N_1$): $N_1 = \sum f_i = 4 + 4 + \alpha + \beta = 8 + \alpha + \beta$
2. Calculate the sum of $f_i x_i$: $\sum f_i x_i = (4 \times 2) + (4 \times 6) + (\alpha \times 8) + (\beta \times 9)$ $\sum f_i x_i = 8 + 24 + 8\alpha + 9\beta = 32 + 8\alpha + 9\beta$
3. Apply the mean formula with $\mu_1 = 6$: $\mu_1 = \frac{\sum f_i x_i}{N_1}$ $6 = \frac{32 + 8\alpha + 9\beta}{8 + \alpha + \beta}$ Multiply both sides by $(8 + \alpha + \beta)$: $6(8 + \alpha + \beta) = 32 + 8\alpha + 9\beta$ $48 + 6\alpha + 6\beta = 32 + 8\alpha + 9\beta$ Rearranging terms to group $\alpha$ and $\beta$: $48 - 32 = 8\alpha - 6\alpha + 9\beta - 6\beta$ $16 = 2\alpha + 3\beta \quad \text{(Equation 1)}$

Step 2: Formulate Equation 2 using the Initial Variance

• What we're doing: We are utilizing the given initial variance ($\sigma_1^2 = 6.8$) along with the initial frequency distribution to derive a second independent equation involving $\alpha$ and $\beta$.
• Why: Having two independent equations with two unknowns ($\alpha$ and $\beta$) allows us to solve for their unique values.

1. Calculate the sum of $f_i x_i^2$: $\sum f_i x_i^2 = (4 \times 2^2) + (4 \times 6^2) + (\alpha \times 8^2) + (\beta \times 9^2)$ $\sum f_i x_i^2 = (4 \times 4) + (4 \times 36) + (64\alpha) + (81\beta)$ $\sum f_i x_i^2 = 16 + 144 + 64\alpha + 81\beta = 160 + 64\alpha + 81\beta$
2. Apply the variance formula with $\sigma_1^2 = 6.8$ and $\mu_1 = 6$: $\sigma_1^2 = \frac{\sum f_i x_i^2}{N_1} - \mu_1^2$ $6.8 = \frac{160 + 64\alpha + 81\beta}{8 + \alpha + \beta} - 6^2$ $6.8 = \frac{160 + 64\alpha + 81\beta}{8 + \alpha + \beta} - 36$ Add 36 to both sides: $6.8 + 36 = \frac{160 + 64\alpha + 81\beta}{8 + \alpha + \beta}$ $42.8 = \frac{160 + 64\alpha + 81\beta}{8 + \alpha + \beta}$ Multiply both sides by $(8 + \alpha + \beta)$: $42.8(8 + \alpha + \beta) = 160 + 64\alpha + 81\beta$ $342.4 + 42.8\alpha + 42.8\beta = 160 + 64\alpha + 81\beta$ Rearranging terms: $342.4 - 160 = 64\alpha - 42.8\alpha + 81\beta - 42.8\beta$ $182.4 = 21.2\alpha + 38.2\beta$ To simplify, multiply by 10 and then divide by 2: $1824 = 212\alpha + 382\beta$ $912 = 106\alpha + 191\beta \quad \text{(Equation 2)}$

Step 3: Solve for $\alpha$ and $\beta$

• What we're doing: We are solving the system of two linear equations obtained in Step 1 and Step 2 to determine the specific values of $\alpha$ and $\beta$.
• Why: Knowing the exact frequencies is crucial for accurately calculating the new mean after the specified data modification.

Our system of equations is:

1. $2\alpha + 3\beta = 16$
2. $106\alpha + 191\beta = 912$

From Equation 1, express $\alpha$ in terms of $\beta$: $2\alpha = 16 - 3\beta \implies \alpha = 8 - \frac{3}{2}\beta$ Substitute this expression for $\alpha$ into Equation 2: $106\left(8 - \frac{3}{2}\beta\right) + 191\beta = 912$ $848 - 159\beta + 191\beta = 912$ $848 + 32\beta = 912$ $32\beta = 912 - 848$ $32\beta = 64$ $\beta = 2$ Now, substitute the value of $\beta = 2$ back into Equation 1 to find $\alpha$: $2\alpha + 3(2) = 16$ $2\alpha + 6 = 16$ $2\alpha = 10$ $\alpha = 5$ Thus, the unknown frequencies are $\alpha=5$ and $\beta=2$.

Step 4: Calculate the New Mean after Data Modification

• What we're doing: We are now applying the specified change to the data ($x_3$ from 8 to 7) and then calculating the mean for this modified frequency distribution using the $\alpha$ and $\beta$ values we just found.
• Why: This is the final requirement of the problem statement.

The new frequency distribution is: $x: \{2, 6, 7, 9\}$ (The value $x_3$ changed from 8 to 7) $f: \{4, 4, 5, 2\}$ (Using $\alpha=5$ and $\beta=2$)

1. Calculate the new total frequency ($N_2$): $N_2 = \sum f_i = 4 + 4 + 5 + 2 = 15$ (Note: Since only an observation value ($x_i$) changed and no frequencies ($f_i$) were altered, the total frequency remains the same as $N_1 = 8+\alpha+\beta = 8+5+2=15$.)
2. Calculate the new sum of $f_i x_i$: $\sum f_i x_i \text{ (new)} = (4 \times 2) + (4 \times 6) + (5 \times 7) + (2 \times 9)$ $\sum f_i x_i \text{ (new)} = 8 + 24 + 35 + 18$ $\sum f_i x_i \text{ (new)} = 85$
3. Calculate the new mean ($\mu_2$): $\mu_2 = \frac{\sum f_i x_i \text{ (new)}}{N_2}$ $\mu_2 = \frac{85}{15}$ Simplify the fraction by dividing both numerator and denominator by 5: $\mu_2 = \frac{17}{3}$

3. Common Mistakes & Tips

• Calculation Errors: Statistics problems, especially those involving multiple steps and equations, are prone to arithmetic errors. Always double-check your sums, products, and algebraic manipulations.
• Choosing the Right Variance Formula: While the definitional formula for variance involves $(x_i - \mu)^2$, the formula $\sigma^2 = \frac{\sum f_i x_i^2}{N} - \mu^2$ is generally more efficient for calculations, particularly when $\mu$ is not an integer or when solving for unknowns.
• Understanding Data Changes: Remember that changing an observation value ($x_i$) affects $\sum f_i x_i$ and $\sum f_i x_i^2$, but typically does not change the total frequency ($N$), unless a frequency ($f_i$) itself is altered or an observation is added/removed.

4. Summary

This problem effectively demonstrates the application of fundamental statistical concepts of mean and variance for frequency distributions. We began by setting up a system of two linear equations derived from the given initial mean and variance. Solving this system allowed us to accurately determine the unknown frequencies $\alpha=5$ and $\beta=2$. With these frequencies, we then incorporated the specified change in data ($x_3$ from 8 to 7) and recalculated the sum of $f_i x_i$. Finally, we divided this new sum by the total frequency to arrive at the new mean.

5. Final Answer

The final answer is $\boxed{\frac{17}{3}}$, which corresponds to option (C).