Key Concepts and Formulas

To accurately solve problems involving the correction of statistical measures, we rely on the fundamental definitions of mean and variance:

1. Mean ($\overline{x}$): The average of a set of $n$ observations $x_1, x_2, \ldots, x_n$. $\overline{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ From this, the sum of observations can be expressed as: $\sum x_i = n \cdot \overline{x}$

2. Variance ($\sigma^2$): A measure of the spread of data points. The most practical computational formula is: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\overline{x})^2$ Rearranging this, the sum of squares of observations can be found: $\sum x_i^2 = n \cdot (\sigma^2 + \overline{x}^2)$

3. Correction Principle: When an incorrect observation ($b$) is replaced by a correct observation ($a$), the sums are adjusted as follows:

   • Corrected Sum of Observations: $(\sum x_i)_{correct} = (\sum x_i)_{incorrect} - b + a$
   • Corrected Sum of Squares of Observations: $(\sum x_i^2)_{correct} = (\sum x_i^2)_{incorrect} - b^2 + a^2$ The number of observations, $n$, remains unchanged in such a replacement scenario.

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Step-by-Step Solution

Let's systematically correct the given statistical measures.

Step 1: Extract Given (Incorrect) Information

• What: Identify all initial parameters provided in the problem statement.
• Why: These values form our baseline before any corrections are applied.
• How:
  • Number of observations ($n$) = 50
  • Incorrect Mean ($\overline{x}_{incorrect}$) = 15
  • Incorrect Standard Deviation ($\sigma_{incorrect}$) = 2
  • From the standard deviation, the Incorrect Variance ($\sigma^2_{incorrect}$) = $2^2 = 4$.

Step 2: Calculate the Incorrect Sum of Observations

• What: Determine the total sum of all observations as they were initially recorded, including the incorrect value.
• Why: This sum is necessary to later subtract the incorrect observation and add the correct one.
• How: Using the formula $\sum x_i = n \cdot \overline{x}$: $(\sum x_i)_{incorrect} = n \cdot \overline{x}_{incorrect}$ $(\sum x_i)_{incorrect} = 50 \cdot 15$ $(\sum x_i)_{incorrect} = 750$

Step 3: Calculate the Incorrect Sum of Squares of Observations

• What: Determine the sum of the squares of all observations as initially recorded.
• Why: This sum is crucial for calculating variance and will be adjusted to find the correct sum of squares.
• How: Using the formula $\sum x_i^2 = n \cdot (\sigma^2 + \overline{x}^2)$: $(\sum x_i^2)_{incorrect} = n \cdot (\sigma^2_{incorrect} + (\overline{x}_{incorrect})^2)$ $(\sum x_i^2)_{incorrect} = 50 \cdot (4 + (15)^2)$ $(\sum x_i^2)_{incorrect} = 50 \cdot (4 + 225)$ $(\sum x_i^2)_{incorrect} = 50 \cdot 229$ $(\sum x_i^2)_{incorrect} = 11450$

Step 4: Establish Relationships for Correct and Incorrect Observations

• What: Define variables for the correct and incorrect observations and form equations based on the given information.
• Why: We need to find the specific values of these observations to properly adjust our sums.
• How: Let $a$ be the correct observation and $b$ be the incorrect observation.
  • Given: "the sum of correct and incorrect observations is 70". $a + b = 70 \quad \text{(Equation 1)}$
  • Given: the correct mean ($\overline{x}_{correct}$) is 16. We can express the correct sum of observations in two ways:
    • Using the correct mean: $(\sum x_i)_{correct} = n \cdot \overline{x}_{correct} = 50 \cdot 16 = 800$.
    • Using the correction principle: $(\sum x_i)_{correct} = (\sum x_i)_{incorrect} - b + a = 750 - b + a$. Equating these two expressions for the correct sum: $800 = 750 - b + a$ Rearranging this gives our second equation: $a - b = 800 - 750$ $a - b = 50 \quad \text{(Equation 2)}$

Step 5: Determine the Values of Correct and Incorrect Observations

• What: Solve the system of linear equations from Step 4.
• Why: This yields the exact values of $a$ and $b$, which are essential for making the final corrections.
• How:
  1. $a + b = 70$
  2. $a - b = 50$ Adding Equation 1 and Equation 2: $(a + b) + (a - b) = 70 + 50$ $2a = 120$ $a = 60$ Substitute $a = 60$ into Equation 1: $60 + b = 70$ $b = 10$ So, the correct observation is 60, and the incorrect observation is 10.

Step 6: Calculate the Correct Sum of Squares of Observations

• What: Adjust the incorrect sum of squares using the values of $a$ and $b$.
• Why: This corrected sum is a direct input for the correct variance calculation.
• How: Using the correction principle $(\sum x_i^2)_{correct} = (\sum x_i^2)_{incorrect} - b^2 + a^2$: $(\sum x_i^2)_{correct} = 11450 - (10)^2 + (60)^2$ $(\sum x_i^2)_{correct} = 11450 - 100 + 3600$ $(\sum x_i^2)_{correct} = 11350 + 3600$ $(\sum x_i^2)_{correct} = 14950$

Step 7: Calculate the Final Correct Variance

• What: Compute the correct variance using the corrected sum of squares, the correct mean, and the number of observations.
• Why: This is the final value requested by the problem.
• How: Using the variance formula $\sigma^2 = \frac{\sum x_i^2}{n} - (\overline{x})^2$: $\sigma^2_{correct} = \frac{(\sum x_i^2)_{correct}}{n} - (\overline{x}_{correct})^2$ $\sigma^2_{correct} = \frac{14950}{50} - (16)^2$ $\sigma^2_{correct} = 299 - 256$ $\sigma^2_{correct} = 43$

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Common Mistakes & Tips

• Squaring vs. Not Squaring: Remember to subtract $b$ and add $a$ for the sum of observations ($\sum x_i$), but subtract $b^2$ and add $a^2$ for the sum of squares of observations ($\sum x_i^2$). This is a common point of error.
• Number of Observations ($n$): In scenarios where an observation is replaced, the number of observations ($n$) remains constant. Only if observations are added or removed without replacement does $n$ change.
• Arithmetic Accuracy: Double-check all calculations, especially squaring numbers and divisions, as a small error can lead to an incorrect final result.

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Summary

This problem required us to correct the mean and variance of a dataset after identifying an incorrect observation. We systematically used the formulas for mean and variance to first calculate the incorrect sums ($\sum x_i$ and $\sum x_i^2$). Then, by utilizing the given correct mean and the relationship between the correct and incorrect observations, we solved for the specific values of the incorrect and correct observations. Finally, these values were used to adjust the sums of observations and sums of squares of observations, leading to the accurate calculation of the correct variance.

The final answer is $\boxed{\text{43}}$, which corresponds to option (A).