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Answer :
The first linear transformation, y = 65 + 1.5x, maintains the original linear relationship between the scores and preserves the relative distances between them, making it more suitable for rescaling the test scores.
To rescale the test scores from the original scale (0-50) to a new scale with a mean of 100 and a standard deviation of 15, we need to apply a linear transformation.
Let's denote the original test scores as x and the rescaled scores as y. We want to find values of a and b such that y = a + bx, where y has a mean of 100 and a standard deviation of 15.
1. Rescaling the mean:
To have a mean of 100, we need to find the value of a. Since the original mean is 35 and the desired mean is 100, we have:
a = desired mean - original mean = 100 - 35 = 65
2. Rescaling the standard deviation:
To have a standard deviation of 15, we need to find the value of b. Since the original standard deviation is 10 and the desired standard deviation is 15, we have:
b = (desired standard deviation) / (original standard deviation) = 15 / 10 = 1.5
Therefore, the linear transformation to rescale the test scores is:
y = 65 + 1.5x
Continuing to the next part of the exercise, there is another linear transformation that can also rescale the scores to have a mean of 100 and a standard deviation of 15. It is given by:
y = 15(x - 35) / 10 + 100
However, this transformation involves multiplying by 15/10 (which is equivalent to 1.5) and adding 100. The reason why this transformation should not be used is that it changes the relative distances between the scores. It stretches the scores vertically and shifts them upward. It may result in a distorted representation of the original scores and can potentially alter the interpretation and comparison of the rescaled scores with other tests.
The first linear transformation, y = 65 + 1.5x, maintains the original linear relationship between the scores and preserves the relative distances between them, making it more suitable for rescaling the test scores.
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