For each student in a certain class, a teacher adjusted the student's test score using the formula y = 0.8x...

1. Translate the problem requirements: We need to understand that we're applying the transformation \(\mathrm{y = 0.8x + 20}\) to every student's score, and we want to find how this affects the spread (standard deviation) of the scores.
2. Understand the effect of linear transformations on standard deviation: When we transform data using \(\mathrm{y = ax + b}\), the standard deviation gets multiplied by \(\mathrm{|a|}\) (the absolute value of the coefficient), while the constant b doesn't affect the spread.
3. Apply the transformation rule: Since our transformation is \(\mathrm{y = 0.8x + 20}\), the new standard deviation will be the original standard deviation multiplied by \(\mathrm{0.8}\).
4. Calculate the final result: Multiply the original standard deviation of 20 by the scaling factor \(\mathrm{0.8}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in plain English. We have a teacher who is adjusting every student's test score using a specific formula: \(\mathrm{y = 0.8x + 20}\). This means that for each student:

• Take their original score (x)
• Multiply it by 0.8 (this makes it smaller)
• Add 20 points to everyone

So if a student originally scored 100, their adjusted score would be: \(\mathrm{0.8(100) + 20 = 80 + 20 = 100}\).
If another student scored 50, their adjusted score would be: \(\mathrm{0.8(50) + 20 = 40 + 20 = 60}\).

We know that the original scores had a standard deviation of 20, and we need to find the standard deviation of the adjusted scores.
Process Skill: TRANSLATE - Converting the formula and standard deviation concept into concrete understanding

2. Understand the effect of linear transformations on standard deviation

Now let's think about what standard deviation measures - it tells us how spread out the scores are from the average.

When we apply the transformation \(\mathrm{y = 0.8x + 20}\) to every score:

• The "+20" part adds the same amount to every single score. This shifts all scores up by 20 points but doesn't change how spread out they are from each other. It's like moving everyone up the same number of steps on a staircase - the gaps between people stay the same.
• The "0.8x" part multiplies every score by 0.8. This is like shrinking the gaps between scores. If two students were originally 10 points apart, after multiplying by 0.8, they'll be \(\mathrm{0.8 × 10 = 8}\) points apart.

So the key insight is: when we transform data using \(\mathrm{y = ax + b}\), the standard deviation gets multiplied by the absolute value of 'a' (the coefficient), while the constant 'b' doesn't affect the spread at all.

3. Apply the transformation rule

Since our transformation is \(\mathrm{y = 0.8x + 20}\), we can see that:

• The coefficient 'a' = 0.8
• The constant 'b' = 20 (this won't affect standard deviation)

Therefore, the new standard deviation will be:
Original standard deviation × |coefficient|
= Original standard deviation × |0.8|
= Original standard deviation × 0.8

4. Calculate the final result

Now we can calculate the final answer:

New standard deviation = Original standard deviation × 0.8
New standard deviation = \(\mathrm{20 × 0.8}\)
New standard deviation = 16

Looking at our answer choices:

1. 12
2. 16 ✓
3. 28
4. 36
5. 40

Final Answer

The standard deviation of the adjusted test scores is 16.

This makes intuitive sense: since we're multiplying each original score by 0.8 (making scores closer together), the spread should also decrease by the same factor of 0.8, giving us \(\mathrm{20 × 0.8 = 16}\).

The answer is B.

Common Faltering Points

Errors while devising the approach

• Misconception about how constants affect standard deviation: Students often think that adding a constant (like +20) to all scores will increase the standard deviation by that same amount. They fail to understand that adding the same value to every data point shifts the entire distribution but doesn't change the spread.
• Confusion about which transformation component affects spread: Students may incorrectly focus on the constant term (+20) rather than the coefficient (0.8) when determining how the transformation affects variability. They might think the standard deviation becomes \(\mathrm{20 + 20 = 40}\) or gets somehow modified by the constant.
• Not recognizing this as a linear transformation problem: Students may attempt to work with individual data points or try to calculate the new mean and standard deviation from scratch, rather than applying the fundamental rule that linear transformations affect standard deviation predictably.

Errors while executing the approach

• Arithmetic error in multiplication: Even when students correctly identify that they need to multiply the original standard deviation by 0.8, they may make a simple calculation error: \(\mathrm{20 × 0.8 = 16}\). Some might incorrectly calculate this as 18 or 14.
• Using addition instead of multiplication: Students might incorrectly add the coefficient to the standard deviation \(\mathrm{(20 + 0.8 = 20.8)}\) or subtract it \(\mathrm{(20 - 0.8 = 19.2)}\) instead of multiplying, showing they don't understand the multiplicative relationship.

Errors while selecting the answer

• Selecting a result that includes the constant term: After calculating correctly that the new standard deviation should be 16, students might second-guess themselves and select an answer that seems to account for the +20 constant, such as 36 (16 + 20) or 40 (20 + 20), thinking they've missed something about how the constant affects the final answer.