Question

the scatterplot shows data set a, which consists of the weights, y, in pounds, of a labrador puppy at various ages, x, in months. the equation of a line of best fit for relationship in data set a can be written as y = -5.0 + 8.7x, where 2 < x < 6. the puppy was weighed again at 9 months old and weighed 52 pounds. data set b consists of all the data points in data set a as well as the data point(9, 52). the equation of a line of best fit for data set b can be written as y = r + sx, where r and s are constants and 2 ≤ x ≤ 9. assuming the equation of the lines of best fit are calculated in the same way, which of the following is the best estimated for the value of s?
a. 5.8
b. 8.7
c. 13.7
d. 17.7

Explanation:

Step1: Analyze Data Set A's Slope

Data set A has line of best fit \( y = -5.0 + 8.7x \), so its slope \( m_A = 8.7 \).

Step2: Analyze Data Set B's Point and Slope Trend

Data set B includes data from A plus the point \( (9, 52) \). We expect the slope \( s \) of B's line of best fit to be less than A's slope (since the new point might flatten the line slightly, but let's check the options). The options are 5.8, 8.7, 13.7, 17.7. Since adding a point outside A's range (\( 2 < x < 6 \)) to A's data (which has slope 8.7) should result in a slope close to but less than 8.7? Wait, no—wait, the original A is \( 2 < x < 6 \), B is \( 2 \leq x \leq 9 \). Let's plug \( x = 9 \) into A's equation: \( y = -5.0 + 8.7(9) = -5 + 78.3 = 73.3 \), but the actual weight at \( x = 9 \) is 52, which is lower. So the line for B should have a smaller slope than 8.7? Wait, no—wait, maybe I misread. Wait, the problem says "the equation of the lines of best fit are calculated in the same way". Wait, maybe the slope of B should be less than 8.7? But the options: 5.8 is less, 8.7 is same, 13.7 and 17.7 are higher. Wait, maybe the scatterplot (not fully visible) shows that as x increases, y increases, but the new point at (9,52) is below the line of A at x=9. So the line of best fit for B should have a slope less than 8.7? Wait, but let's check the options. Wait, maybe I made a mistake. Wait, the line of best fit for A is \( y = -5.0 + 8.7x \). For data set B, which includes A's points and (9,52), we can think about the slope. Let's consider the trend: if the new point is below the A's line at x=9, the slope of B's line should be less than 8.7. Among the options, 5.8 is less, 8.7 is same, 13.7 and 17.7 are higher. So the best estimate is 5.8? Wait, no—wait, maybe the scatterplot (the grid) shows points increasing, but the new point is at (9,52). Let's check the y-axis: the grid has y from, say, 0 to 10? Wait, no, the grid on the right: x-axis (age) and y-axis (weight). Wait, maybe the original A's line has slope 8.7, and B's line, including (9,52), should have a slope less than 8.7, so 5.8? But wait, maybe I messed up. Wait, the options are a.5.8, b.8.7, c.13.7, d.17.7. Since the new point is at x=9, y=52, and A's line at x=9 is y=-5 + 8.7*9=73.3, which is higher than 52, so the line of best fit for B will have a smaller slope than 8.7. So the only option less than 8.7 is 5.8.

Answer:

a. 5.8