Question

many local water companies encourage their customers to use devices that limit water use to help reduce their water bills. the table provides information about the relationship between the number of water conservation devices and monthly water bills for a random selection of 9 homes that receive their water from a local water company. use technology to calculate the equation of the least - squares regression line relating ( y = ) water bill to ( x = ) number of water conservation devices. ( \bigcirchat{y}=-36.884 - 4.386x ) ( \bigcirchat{y}=36.884 - 4.386x ) ( \bigcirchat{y}=36.884 + 4.386x ) ( \bigcirchat{y}=-4.386 + 36.884x ) ( \bigcirc ) option a ( \bigcirc ) option b ( \bigcirc ) option c ( \bigcirc ) option d

Explanation:

Step1: Analyze the general form of regression line

The general form of a least - squares regression line is $\hat{y}=a + bx$, where $a$ is the y - intercept and $b$ is the slope.

Step2: Consider the relationship between $x$ (number of devices) and $y$ (water bill)

As the number of water conservation devices ($x$) increases, the water bill ($y$) decreases. So the slope $b$ should be negative.

Step3: Analyze the y - intercept

When $x = 0$ (no devices), we expect a positive water bill.

For the equation $\hat{y}=a+bx$:

• If we substitute $x = 0$ into $\hat{y}=-36.884-4.386x$, we get $\hat{y}=-36.884$ (not possible as water bill can't be negative when $x = 0$).
• If we substitute $x = 0$ into $\hat{y}=36.884-4.386x$, we get $\hat{y}=36.884$ (reasonable as a starting water bill when $x = 0$)
• If we substitute $x = 0$ into $\hat{y}=36.884 + 4.386x$, we get $\hat{y}=36.884$, but the slope is positive (contradicts the negative relationship between $x$ and $y$)
• If we substitute $x = 0$ into $\hat{y}=-4.386+36.884x$, we get $\hat{y}=-4.386$ (not possible as water bill can't be negative when $x = 0$)

Answer:

Option B