Using the concepts and properties of mean and standard deviation, it is found that:
- Each test score has to be multiplied by 4 and increased by 27.
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- The mean of a data-set is the sum of all values in the data-set divided by the number of values.
- If all values are increased by x, the mean will also be increased by x.
- If all values are multiplied by x, the mean will also be multiplied by x.
- The standard deviation of a data-set is the square root of the sum of the differences squared of each value and the mean, divided by the number of values.
- If all values are increased by x, the standard deviation remains constant.
- If all values are multiplied by x, the standard deviation is also multiplied by x.
In this problem:
- First, we multiply, to find the desired standard deviation, then we add, to find the desired mean.
- Standard deviation of 3 we want to be 12, thus, we multiply each test score by 4, as [tex]\frac{12}{3} = 4[/tex].
- Multiplying by 4, the standard deviation will already be 12, but the mean will be 48.
- We want a mean of 75, thus, we have to increase each test score by 27, as 75 - 48 = 27.
A similar problem is given at https://brainly.com/question/20640860