Final answer:
The new variance when the marks out of 75 with a standard deviation of 9 are raised to a total of 100 is 144. This is found by squaring the scaling factor, which is 4/3, and multiplying it by the original variance of 81.
Step-by-step explanation:
The question asks to calculate the new variance when the marks of some students, which were originally out of a total of 75 with a standard deviation of 9, are raised to a maximum of 100. To find the new variance, we need to understand the relation between the variance and scaling of data.
When all values of a dataset are multiplied by a constant, the variance is multiplied by the square of that constant. In this case, to adjust the marks to a total of 100, a scaling factor is applied. Since the original total is 75, and the new total is 100, the scaling factor is \(\frac{100}{75} = \frac{4}{3}\).
The original variance, which is the square of the standard deviation, is \(9^2 = 81\). Applying the scaling factor to the variance, we get the new variance: \(81 \times \left(\frac{4}{3}\right)^2 = 81 \times \frac{16}{9} = 144\).
Therefore, the new variance is 144, which is option c.