Question

which of the following equations could be the regression line for the scatterplot shown below?
select the correct answer
y = (1/3)x + 50
y = (1/3)x - 300
y = (9/2)x + 50
y = (9/2)x - 300

Explanation:

Step1: Analyze the slope and intercept trend

The scatterplot shows a positive correlation (as x increases, y increases). Let's check the y - intercept and slope. When x is around 90 (from the x - axis), y is around 60 - 70. Let's test the equations.

Step2: Test the y - intercept and slope for each equation

- For \(y=\frac{1}{3}x + 50\): The slope is small, and the line would be less steep.
- For \(y=\frac{1}{3}x-300\): When x = 90, \(y=\frac{1}{3}\times90 - 300=30 - 300=- 270\), which is way below the scatterplot's y - values.
- For \(y=\frac{9}{2}x + 50\): The slope is very large, the line would be too steep.
- For \(y=\frac{9}{2}x-300\): Let's plug x = 90, \(y=\frac{9}{2}\times90-300 = 405 - 300 = 105\)? No, wait, let's check x = 100. Wait, maybe better to check the general trend. Wait, actually, when x is around 90, let's see the first few points. Wait, maybe I made a mistake. Wait, let's re - evaluate.

Wait, the scatterplot has x - values starting from around 90 (maybe 92, 93 etc.) and y - values around 60 - 70. Let's take x = 90.

For \(y=\frac{1}{3}x-300\): \(y = 30-300=-270\) (wrong).

For \(y=\frac{1}{3}x + 50\): \(y=\frac{1}{3}\times90 + 50=30 + 50 = 80\), which is close to the middle - range y - values.

Wait, no, maybe I messed up the x - axis. Wait, the x - axis labels: 90, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105.

Let's take x = 90, y should be around 60? Wait, no, the first point is around x = 92, y = 60. Let's plug x = 92 into \(y=\frac{1}{3}x-300\): \(y=\frac{92}{3}-300\approx30.67 - 300=-269.33\) (wrong).

Plug x = 92 into \(y=\frac{1}{3}x + 50\): \(y=\frac{92}{3}+50\approx30.67 + 50 = 80.67\), which is higher than the first point's y - value (around 60). Wait, maybe I misread the x - axis. Wait, maybe the x - axis is from 90 to 105, and the y - axis from 40 to 120.

Wait, let's take x = 100. For \(y=\frac{1}{3}x-300\): \(y=\frac{100}{3}-300\approx33.33 - 300=-266.67\) (wrong).

For \(y=\frac{1}{3}x + 50\): \(y=\frac{100}{3}+50\approx33.33 + 50 = 83.33\), which is in the range of the middle y - values.

For \(y=\frac{9}{2}x-300\): \(y = 4.5\times100-300 = 450 - 300 = 150\) (way above the scatterplot's max y - value of ~110).

For \(y=\frac{9}{2}x + 50\): \(y = 4.5\times100+50 = 500\) (way above).

Wait, so the only equation that makes sense in terms of y - values (not going negative and not too steep) is \(y=\frac{1}{3}x + 50\)? No, wait, no, I think I made a mistake. Wait, the correct equation should have a positive slope (since the scatterplot has a positive correlation) and a y - intercept that makes sense.

Wait, let's check the second equation \(y=\frac{1}{3}x-300\): when x = 100, \(y=\frac{100}{3}-300\approx - 266\) (wrong). The third equation \(y=\frac{9}{2}x + 50\): slope is too big. The fourth equation \(y=\frac{9}{2}x-300\): when x = 100, \(y = 450-300 = 150\) (too big). The first equation \(y=\frac{1}{3}x + 50\): when x = 90, \(y = 30 + 50 = 80\), which is in the range of the scatterplot's y - values (which go from ~60 to ~110).

Wait, maybe the correct answer is \(y=\frac{1}{3}x + 50\)? No, wait, no, I think I messed up. Wait, let's re - calculate.

Wait, the scatterplot has a positive correlation, so the slope is positive. Now, let's check the y - intercept. The line should pass through the "middle" of the points. Let's take two points: say (90, 60) and (100, 90). The slope between them is \(\frac{90 - 60}{100 - 90}=\frac{30}{10}=3\), but the options have slopes \(\frac{1}{3}\) and \(\frac{9}{2}=4.5\). Wait, maybe my point selection is wrong.

Wait, the options are \(\frac{1}{3}\) (slope ~0.33) and \(\frac{9}{2}=4.5\) (slope 4.5). The scatterplot's points are increasing, but not very steeply, so the slope should be small. So between \(\frac{1}{3}x + 50\) and \(\frac{1}{3}x-300\). The y - intercept for \(\frac{1}{3}x-300\) is negative, which would mean at x = 0, y is negative, but our x - values start at 90, so when x = 90, y=\(\frac{1}{3}\times90-300 = 30 - 300=-270\), which is way below the scatterplot. The \(y=\frac{1}{3}x + 50\) at x = 90 gives y = 80, which is in the range of the scatterplot's y - values (which are from ~60 to ~110). So the correct equation should be \(y=\frac{1}{3}x + 50\)? Wait, no, wait the other option \(y=\frac{9}{2}x-300\): when x = 100, \(y = 450-300 = 150\), which is above the scatterplot's max y (110). \(y=\frac{9}{2}x + 50\) at x = 100 is \(450 + 50 = 500\), way too high.

So between the two \(\frac{1}{3}\) slope equations: \(y=\frac{1}{3}x + 50\) has a positive y - intercept, \(y=\frac{1}{3}x-300\) has a negative y - intercept. Since the scatterplot's y - values are positive (above 40), the y - intercept should be positive. So the equation is \(y=\frac{1}{3}x + 50\). Wait, but when x = 90, \(y=\frac{1}{3}\times90+50 = 30 + 50 = 80\), and the points at x = 90 are around 60 - 70, but maybe the "middle" is around 80. So I think the correct answer is \(y=\frac{1}{3}x + 50\). Wait, no, wait the other option \(y=\frac{1}{3}x-300\) is wrong because of the negative y - intercept. So the correct equation is \(y=\frac{1}{3}x + 50\).

Wait, no, I think I made a mistake. Let's check the x - axis again. The x - axis starts at 90, and the y - axis starts at 40. So when x = 90, the line \(y=\frac{1}{3}x-300\) would be at y = - 270, which is way below the graph. The line \(y=\frac{1}{3}x + 50\) at x = 90 is at y = 80, which is within the y - range of the graph (40 - 120). So the correct equation is \(y=\frac{1}{3}x + 50\).

Answer:

\(y=\frac{1}{3}x + 50\)