Question

a large set of test scores has mean 60 and standard deviation 18. suppose each score is doubled, and then 5 is subtracted from the result.
(a) find the mean of the new scores. (enter an integer.)
(b) find the standard deviation of the new score (enter an integer.)

Explanation:

Step1: Recall the formula for the mean of a transformed variable

If \(X\) is a random variable with mean \(\mu_X\), and \(Y = aX + b\) (where \(a\) and \(b\) are constants), then \(\mu_Y=a\mu_X + b\). Here \(a = 2\) and \(b=- 5\), and \(\mu_X = 60\).

$$ \mu_Y=2\times60-5 $$

Step2: Calculate the new mean

$$ \mu_Y = 120 - 5=115 $$

Step3: Recall the formula for the standard deviation of a transformed variable

If \(X\) is a random variable with standard deviation \(\sigma_X\), and \(Y=aX + b\) (where \(a\) and \(b\) are constants), then \(\sigma_Y=\vert a\vert\sigma_X\). Here \(a = 2\) and \(b =-5\), and \(\sigma_X=18\).

$$ \sigma_Y=\vert2\vert\times18 $$

Step4: Calculate the new standard deviation

$$ \sigma_Y=36 $$

Answer:

(a) \(115\)
(b) \(36\)