Question

the scatter plot shows the relationship between the test score, y, out of 100, and the total number of minutes, x, students spend studying for a test. which function best represents the linear model for the data in the scatter plot? options: y = 0.5x + 10, y = 0.5x + 50, y = 2x + 10, y = 2x + 50

Explanation:

Step1: Analyze the scatter plot trend

The scatter plot shows a positive linear relationship, so the slope should be positive. Let's check the options: all have positive slopes, so we need to find the correct slope and y - intercept.

Step2: Estimate two points

From the scatter plot, let's assume two points. For example, when \(x = 15\) (approximate), \(y\approx17.5\); when \(x = 30\), \(y\approx25\). Let's use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Taking \((x_1,y_1)=(15,17.5)\) and \((x_2,y_2)=(30,25)\), then \(m=\frac{25 - 17.5}{30 - 15}=\frac{7.5}{15}=0.5\).

Step3: Check the y - intercept

Now, use the point - slope form \(y - y_1=m(x - x_1)\). Using \(m = 0.5\) and the point \((15,17.5)\), we get \(y-17.5 = 0.5(x - 15)\). Expanding, \(y-17.5=0.5x - 7.5\), so \(y = 0.5x+10\). Let's check another point. If \(x = 45\), \(y=0.5\times45 + 10=22.5 + 10 = 32.5\), which is close to the trend of the scatter plot. The other options: for \(y = 2x+10\), when \(x = 15\), \(y = 40\) (too high); for \(y=0.5x + 50\), when \(x = 15\), \(y=57.5\) (too high); for \(y = 2x+50\), when \(x = 15\), \(y = 80\) (too high).

Answer:

\(y = 0.5x+10\) (the first option among the given options, assuming the options are \(y = 0.5x + 10\), \(y=0.5x + 50\), \(y = 2x+10\), \(y = 2x+50\) in order)