The Correct Answer and Explanation is:
Of course. The image text is incomplete and cuts off after “standard deviation of”. For this solution, we will assume the original standard deviation was 3.
Correct Answer:
Ms. Martin should first multiply each quiz score by 4 and then add 27 to the result. The transformation is: New Score = 4 * (Old Score) + 27.
Explanation
This problem requires a linear transformation to adjust the statistical properties of a dataset. A linear transformation takes the form Y = aX + b, where X is the original score, Y is the new score, ‘a’ is a multiplicative factor, and ‘b’ is an additive factor.
The key to solving this is understanding how these two factors affect the mean and standard deviation.
- Effect on Standard Deviation: The standard deviation, which measures the spread of the data, is only affected by multiplication or division. Adding or subtracting a constant shifts the entire dataset but does not change its spread. Therefore, the new standard deviation is calculated as: New SD = |a| * Old SD.
- Effect on Mean: The mean, which measures the center of the data, is affected by both multiplication and addition. The new mean is calculated as: New Mean = a * (Old Mean) + b.
We can solve for the transformation in two steps.
Step 1: Find the multiplicative factor ‘a’ to fix the standard deviation.
We need to transform the original standard deviation (assumed to be 3) into the new standard deviation of 12.
Using the formula:
12 = a * 3
a = 12 / 3
a = 4
This tells us that every original score must be multiplied by 4.
Step 2: Find the additive factor ‘b’ to fix the mean.
First, we apply the multiplication from Step 1 to the original mean:
4 * (Original Mean) = 4 * 12 = 48.
After multiplying, our scores have a mean of 48 and a standard deviation of 12. The standard deviation is now correct, but the mean is not. We need a mean of 75. To get from 48 to 75, we must add a constant.
b = New Mean – Intermediate Mean
b = 75 – 48
b = 27
This tells us we must add 27 after multiplying.
Combining these steps gives the complete transformation: multiply each score by 4, then add 27.
