Key Concepts and Formulas

• Linear Transformation of Data: When 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 ($a \ne 0$).
• Effect on Mean (Change of Origin and Scale): The new mean, $\bar{y}$, is related to the old mean, $\bar{x}$, by the same linear transformation: $\bar{y} = a\bar{x} + b$
• Effect on Standard Deviation (Change of Scale Only): The new standard deviation, $\sigma_y$, is related to the old standard deviation, $\sigma_x$, only by the absolute value of the scaling factor: $\sigma_y = |a|\sigma_x$ Note that the constant $b$ (shifting factor) does not affect the standard deviation, as it only shifts the entire distribution without changing its spread.

Step-by-Step Solution

Let's break down the problem using the given information and the established statistical rules.

Given Information:

• Original mean ($\bar{x}$) = 20
• Original standard deviation ($\sigma_x$) = 2
• Transformation: Each observation $x_i$ is multiplied by $p$ and then reduced by $q$. So, the new observation $y_i$ is given by: $y_i = px_i - q$. Comparing this to the general form $y_i = ax_i + b$, we identify:
  • Scaling factor ($a$) = $p$
  • Shifting factor ($b$) = $-q$
• New mean ($\bar{y}$) = half of original mean = $\frac{1}{2} \bar{x} = \frac{1}{2}(20) = 10$
• New standard deviation ($\sigma_y$) = half of original standard deviation = $\frac{1}{2} \sigma_x = \frac{1}{2}(2) = 1$
• Constraints: $p \ne 0$ and $q \ne 0$.

Step 1: Apply the Transformation Rule for Standard Deviation

We use the formula for the transformed standard deviation: $\sigma_y = |a|\sigma_x$. Substitute the identified value $a=p$: $\sigma_y = |p|\sigma_x$ Now, substitute the given numerical values for $\sigma_x$ and $\sigma_y$: $1 = |p|(2)$ To find $|p|$, we divide by 2: $|p| = \frac{1}{2}$ This equation tells us that $p$ can be either $\frac{1}{2}$ or $-\frac{1}{2}$. We will explore both possibilities in the next step.

Step 2: Apply the Transformation Rule for the Mean

Next, we use the formula for the transformed mean: $\bar{y} = a\bar{x} + b$. Substitute the identified values $a=p$ and $b=-q$: $\bar{y} = p\bar{x} - q$ Now, substitute the given numerical values for $\bar{x}$ and $\bar{y}$: $10 = p(20) - q$ $10 = 20p - q \quad \ldots(1)$ This equation relates $p$ and $q$. We will use the possible values of $p$ from Step 1 to solve for $q$.

Step 3: Solve for $q$ using the Possible Values of $p$

We have two possible values for $p$ from Step 1 ($p = \frac{1}{2}$ or $p = -\frac{1}{2}$).

Case 1: Assume $p = \frac{1}{2}$ Substitute $p = \frac{1}{2}$ into Equation (1): $10 = 20\left(\frac{1}{2}\right) - q$ $10 = 10 - q$ Solving for $q$: $q = 10 - 10$ $q = 0$ However, the problem statement explicitly states that $q \ne 0$. Therefore, this case where $p = \frac{1}{2}$ is invalid.

Case 2: Assume $p = -\frac{1}{2}$ Substitute $p = -\frac{1}{2}$ into Equation (1): $10 = 20\left(-\frac{1}{2}\right) - q$ $10 = -10 - q$ Solving for $q$: $q = -10 - 10$ $q = -20$ This value of $q = -20$ satisfies the condition $q \ne 0$. Also, $p = -\frac{1}{2}$ satisfies $p \ne 0$. Both conditions are met in this case.

Step 4: Verify the Solution

Let's check if $p = -\frac{1}{2}$ and $q = -20$ satisfy all the problem conditions:

• Original mean $\bar{x} = 20$. New mean $\bar{y} = p\bar{x} - q = (-\frac{1}{2})(20) - (-20) = -10 + 20 = 10$. This is indeed half of the original mean.
• Original s.d. $\sigma_x = 2$. New s.d. $\sigma_y = |p|\sigma_x = |-\frac{1}{2}|(2) = \frac{1}{2}(2) = 1$. This is indeed half of the original s.d. All conditions are consistent with the calculated values.

Common Mistakes & Tips

• Absolute Value for Standard Deviation: Always remember to use the absolute value of the scaling factor ($|a|$) when transforming the standard deviation. A common error is to use $a$ directly, which can lead to incorrect signs or values for $p$.
• Shifting Does Not Affect Standard Deviation: Understand that adding or subtracting a constant to all observations (the 'b' term) shifts the entire distribution but does not change its spread, and thus has no effect on the standard deviation.
• Check Constraints: Always verify that your final values for $p$ and $q$ satisfy any given constraints, such as $p \ne 0$ and $q \ne 0$. This step was crucial in eliminating one of the possibilities for $p$.

Summary

This problem requires a clear understanding of how linear transformations affect the mean and standard deviation of a dataset. By applying the specific rules that the mean is affected by both scaling and shifting, while the standard deviation is only affected by the absolute value of the scaling factor, we set up a system of equations. Solving these equations and considering the given constraints led us to the unique values for $p$ and $q$. We found that $p = -\frac{1}{2}$ and $q = -20$.

The final answer is $\boxed{-20}$, which corresponds to option (B).