Question 19

Explanation

We will determine the values of $a$ and $b$ based on how linear transformations affect the mean and range of a dataset.

Let the initial mean be $\bar{x}_1 = 5$ and the initial range be $R_1 = 4$. The data transformation involves multiplying each value by $a$ and then adding $b$.

Analysis of Range Change

The range is the difference between the maximum and minimum values. Adding a constant $b$ shifts all data points equally, so the difference remains unchanged. However, multiplying by $a$ scales the range. Since the range is always positive:

Given the new range $R_2 = 12$ and the old range $R_1 = 4$. Assuming $a$ is positive in this context:

Analysis of Mean Change

The mean is affected by both multiplication and addition. The transformation formula for the mean is:

Substitute the known values ($\bar{x}_2 = 19$, $\bar{x}_1 = 5$, and $a = 3$):

Calculating the Final Value

The question asks for the value of $3a - b$:

Thus, the value of $3a - b$ is $5$.