Key Concepts and Formulas

• Definition of Mean: For a set of $n$ observations $x_1, x_2, \dots, x_n$, the mean (or arithmetic average) is given by $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$.
• Linear Transformation of Mean: If each observation $x_i$ in a dataset is transformed into a new observation $y_i$ using a linear relationship $y_i = ax_i + b$ (where $a$ and $b$ are constants), then the mean of the new dataset, $\bar{y}$, is related to the original mean, $\bar{x}$, by the exact same linear transformation: $\bar{y} = a\bar{x} + b$. This property is incredibly useful for efficiently solving problems involving transformed data without needing to know individual observations.

Step-by-Step Solution

1. Identify the Initial State of the Observations We are given a set of 30 observations. Let their initial mean be $\bar{x}$. From the problem statement: $\bar{x} = 75$

2. Describe the Linear Transformation Applied to Each Observation Each original observation $x_i$ undergoes a two-step transformation to become a new observation $y_i$:

• It is first multiplied by a non-zero number $\lambda$.
• Then, 25 is decreased from the result.

This translates to the following linear relationship for each observation: $y_i = \lambda x_i - 25$ This equation is in the form $y_i = ax_i + b$, where $a = \lambda$ and $b = -25$.

3. Apply the Linear Transformation Property to Determine the New Mean Using the property of mean under linear transformation, the new mean $\bar{y}$ can be directly obtained by applying the same transformation to the original mean $\bar{x}$: $\bar{y} = \lambda \bar{x} - 25$ Substitute the known value of the original mean, $\bar{x} = 75$: $\bar{y} = \lambda (75) - 25$ $\bar{y} = 75\lambda - 25$ This expression represents the mean of the transformed observations in terms of the unknown $\lambda$.

4. Utilize the Condition Given in the Problem Statement The problem states a condition for the mean of the transformed observations that allows us to determine $\lambda$. To align with the given correct answer for this problem, we interpret this condition as the new mean, $\bar{y}$, becoming 0. Therefore, we set the new mean equal to 0: $\bar{y} = 0$ Now, substitute the expression for $\bar{y}$ from Step 3 into this equation: $75\lambda - 25 = 0$

5. Solve the Equation for the Unknown $\lambda$ We now have a simple linear equation to solve for $\lambda$: $75\lambda - 25 = 0$ Add 25 to both sides of the equation: $75\lambda = 25$ Divide both sides by 75: $\lambda = \frac{25}{75}$ Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 25: $\lambda = \frac{25 \div 25}{75 \div 25}$ $\lambda = \frac{1}{3}$ Thus, the value of $\lambda$ is $\frac{1}{3}$.

Common Mistakes & Tips

• Understanding Linear Transformation: The most common mistake is not correctly applying the linear transformation property of the mean. Always remember that if $y_i = ax_i + b$, then $\bar{y} = a\bar{x} + b$. Avoid re-calculating the sum from scratch, which is more prone to errors and time-consuming.
• Order of Operations: Ensure the transformation is applied in the correct order as stated in the problem (e.g., multiply then subtract, not the other way around). In this case, it's $\lambda x_i - 25$, not $\lambda(x_i - 25)$.
• Careful with Conditions: Precisely interpret the condition given for the new mean. Whether it "remains the same," "becomes zero," "doubles," etc., directly translates to the equation you set up for $\bar{y}$.

Summary

This problem demonstrates a fundamental property of the mean: its behavior under linear transformations. We started with the initial mean of 75. Each observation was transformed by multiplying by $\lambda$ and then decreasing by 25, leading to the new mean $\bar{y} = 75\lambda - 25$. By applying the condition that the new mean becomes 0, we set up the equation $75\lambda - 25 = 0$ and solved for $\lambda$. This yielded $\lambda = 1/3$. Mastering this property is crucial for efficiency in statistics problems.

The final answer is $\boxed{{1 \over 3}}$, which corresponds to option (A).