Key Concepts and Formulas

For a set of $n$ observations $x_1, x_2, \ldots, x_n$:

1. Arithmetic Mean ($\overline{x}$): The sum of all observations divided by the number of observations. $\overline{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

2. Variance ($\sigma^2$): The average of the squared differences from the mean. $\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \overline{x})^2}{n}$

3. Properties of Mean and Variance under Linear Transformation: If each observation $x_i$ is transformed to $y_i = ax_i + b$ (where $a$ and $b$ are constants), then:

   • The new mean ($\overline{y}$) is $\overline{y} = a\overline{x} + b$.
   • The new variance ($\sigma_y^2$) is $\sigma_y^2 = a^2 \sigma^2$.

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Step-by-Step Solution

Let the original set of $n$ observations be $x_1, x_2, \ldots, x_n$. Given that their arithmetic mean is $\overline{x}$ and their variance is $\sigma^2$.

We are now considering a new set of observations: $2x_1, 2x_2, \ldots, 2x_n$. Let's denote these new observations as $y_i$, so $y_i = 2x_i$. This is a linear transformation where $a=2$ and $b=0$.

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1. Analyzing Statement 2: Arithmetic Mean of $2x_1, 2x_2, \ldots, 2x_n$

Statement 2 says: Arithmetic mean of $2x_1, 2x_2, \ldots, 2x_n$ is $4\overline{x}$.

• Step 1: Calculate the new arithmetic mean ($\overline{y}$) using its definition. The arithmetic mean of the new observations $y_1, y_2, \ldots, y_n$ (where $y_i = 2x_i$) is given by: $\overline{y} = \frac{\sum_{i=1}^{n} y_i}{n}$ Why this step? We start with the fundamental definition to derive the new mean.

• Step 2: Substitute $y_i = 2x_i$ and simplify. $\overline{y} = \frac{2x_1 + 2x_2 + \ldots + 2x_n}{n}$ Factor out the common multiplier '2' from the numerator: $\overline{y} = \frac{2(x_1 + x_2 + \ldots + x_n)}{n}$ Why this step? Factoring out the constant helps to reveal the relationship with the original mean.

• Step 3: Relate to the original mean ($\overline{x}$). We know that the original arithmetic mean is $\overline{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}$. Substitute this into our expression for $\overline{y}$: $\overline{y} = 2 \left( \frac{\sum_{i=1}^{n} x_i}{n} \right) = 2\overline{x}$ Why this step? This directly shows how the mean transforms under multiplication by a constant.

• Step 4: Compare with Statement 2. Our calculated new mean is $2\overline{x}$. Statement 2 claims the mean is $4\overline{x}$. Since $2\overline{x} \neq 4\overline{x}$ (unless $\overline{x}=0$, which is not generally true), Statement 2 is false.

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2. Analyzing Statement 1: Variance of $2x_1, 2x_2, \ldots, 2x_n$

Statement 1 says: Variance of $2x_1, 2x_2, \ldots, 2x_n$ is $4\sigma^2$.

• Step 1: Calculate the new variance ($\sigma_y^2$) using its definition. The variance of the new observations $y_1, y_2, \ldots, y_n$ is: $\sigma_y^2 = \frac{\sum_{i=1}^{n} (y_i - \overline{y})^2}{n}$ Why this step? We use the fundamental definition of variance to derive its transformation.

• Step 2: Substitute $y_i = 2x_i$ and the previously calculated $\overline{y} = 2\overline{x}$ into the formula. $\sigma_y^2 = \frac{\sum_{i=1}^{n} (2x_i - 2\overline{x})^2}{n}$ Why this step? This substitutes the transformed values and their new mean into the variance definition.

• Step 3: Simplify the expression inside the summation. Factor out '2' from the term $(2x_i - 2\overline{x})$: $\sigma_y^2 = \frac{\sum_{i=1}^{n} (2(x_i - \overline{x}))^2}{n}$ Now, square the term $2(x_i - \overline{x})$: $\sigma_y^2 = \frac{\sum_{i=1}^{n} 4(x_i - \overline{x})^2}{n}$ Why this step? This simplification is crucial to isolate the original variance term.

• Step 4: Factor out the constant from the summation. The constant '4' can be taken out of the summation: $\sigma_y^2 = 4 \left( \frac{\sum_{i=1}^{n} (x_i - \overline{x})^2}{n} \right)$ Why this step? This prepares the expression to be directly compared with the original variance.

• Step 5: Relate to the original variance ($\sigma^2$). We know that the original variance is $\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \overline{x})^2}{n}$. Substituting this into our expression for $\sigma_y^2$: $\sigma_y^2 = 4\sigma^2$ Why this step? This confirms how the variance scales when observations are multiplied by a constant.

• Step 6: Compare with Statement 1. Our calculated new variance is $4\sigma^2$. Statement 1 claims the variance is $4\sigma^2$. Thus, Statement 1 is true.

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3. Conclusion and Option Selection

• Statement 1 is True.
• Statement 2 is False.

This combination corresponds to option (D).

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Common Mistakes & Tips

• Scaling Factors: Remember that if observations are multiplied by a constant 'a', the mean is multiplied by 'a', but the variance is multiplied by $a^2$. A common error is to multiply variance by 'a' instead of $a^2$.
• Effect of Addition/Subtraction: Adding or subtracting a constant 'b' to all observations changes the mean by 'b' but does not affect the variance. This is because the differences from the mean, $(x_i - \overline{x})$, remain unchanged.
• Always Revert to Definitions: If you are unsure about a property, always go back to the fundamental definitions of mean and variance and derive the transformation step-by-step.

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Summary

We systematically analyzed both statements using the definitions and properties of arithmetic mean and variance under linear transformation. For the new observations $2x_i$, the new mean was found to be $2\overline{x}$, making Statement 2 (mean is $4\overline{x}$) false. The new variance was found to be $4\sigma^2$, making Statement 1 (variance is $4\sigma^2$) true. Therefore, Statement 1 is true and Statement 2 is false.

The final answer is \boxed{D}