Question

the scores on ms. martins statistics quiz had a mean of 12 and a standard deviation of 3. ms. martin wants to transform the scores to have a mean of 75 and a standard deviation of 12. what transformations should she apply to each test score? new score = 4(old score) + 27 new score = 4(old score) + 27 new score = 4(old score) new score = 6(old score) + 3

Explanation:

Step1: Recall the rules for mean and standard deviation of linear transformations

For a linear transformation \(y = a x + b\), the mean \(\mu_y=a\mu_x + b\) and the standard deviation \(\sigma_y=\vert a\vert\sigma_x\).

Step2: Find the value of \(a\) using the standard - deviation formula

We know that \(\sigma_x = 3\) and \(\sigma_y=12\). Using the formula \(\sigma_y=\vert a\vert\sigma_x\), we substitute the values: \(12=\vert a\vert\times3\). Solving for \(a\), we get \(a = 4\) (since we are dealing with a scaling factor for scores, we can assume \(a>0\)).

Step3: Find the value of \(b\) using the mean formula

We know that \(\mu_x = 12\), \(a = 4\), and \(\mu_y=75\). Using the formula \(\mu_y=a\mu_x + b\), we substitute the values: \(75=4\times12 + b\).

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Answer:

New score \(=4(\text{old score})+27\)