Question

a number line with weights of marked points for a weighted average is shown. what statement is true? a) the weighted average is -0.5. b) if the weight of -2 is increased, the weighted average would increase. c) if the weight of 1 is increased, the weighted average would increase. d) if the weight of -2 is decreased to 1 and the weight of 1 is decreased to 3, the weighted average would decrease.

Explanation:

Step1: Recall weighted - average formula

The weighted - average formula is $\bar{x}=\frac{\sum_{i = 1}^{n}w_ix_i}{\sum_{i = 1}^{n}w_i}$, where $w_i$ is the weight and $x_i$ is the value. Here, $x_1=-2$, $w_1 = 2$, $x_2 = 1$, $w_2=6$. Then $\bar{x}=\frac{2\times(-2)+6\times1}{2 + 6}=\frac{-4 + 6}{8}=\frac{2}{8}=0.25$.

Step2: Analyze option B

If the weight of $-2$ (a negative number) is increased, the numerator $\sum_{i = 1}^{n}w_ix_i$ will become more negative (since $x_1=-2$), and the weighted - average will decrease.

Step3: Analyze option C

If the weight of $1$ (a positive number) is increased, the numerator $\sum_{i = 1}^{n}w_ix_i$ will become more positive. Since the denominator $\sum_{i = 1}^{n}w_i$ also increases, but the positive contribution to the numerator grows, the weighted - average will increase.

Step4: Analyze option D

If the weight of $-2$ is decreased to $1$ and the weight of $1$ is decreased to $3$, the new weighted - average is $\bar{x}=\frac{1\times(-2)+3\times1}{1 + 3}=\frac{-2 + 3}{4}=\frac{1}{4}=0.25$, which is the same as the original weighted - average.

Answer:

C. If the weight of 1 is increased, the weighted average would increase.