Question

Xavier created a scatterplot and a trend line for data that he collected by comparing the number of minutes he spent playing a game and the number of levels he passed.

Video Game
A graph has minutes played on the x-axis and levels passed on the y-axis. A line goes through points (10, 4) and (30, 14).

According to the graph, which combination of minutes played and levels passed seem most likely to be possible?
2 minutes played and 3 levels passed
17 minutes played and 35 levels passed
20 minutes played and 9 levels passed
50 minutes played and 19 levels passed

Answer 1

To determine which combination of minutes played and levels passed is most likely, we need to analyze the trend line represented by the points (10, 4) and (30, 14).

1. **Find the slope of the trend line**:
The slope \( m \) is calculated as follows:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{14 - 4}{30 - 10} = \frac{10}{20} = 0.5
\]

Therefore, the slope of the trend line is 0.5. This means for every minute played, approximately 0.5 levels are passed.

2. **Equation of the trend line**:
Using point-slope form, we can write the equation of the trend line starting from the point (10, 4).

\[
y - 4 = 0.5(x - 10)
\]

Simplifying this gives:

\[
y = 0.5x + 4 - 5 = 0.5x - 1
\]

3. **Evaluate each combination**:
We can substitute each combination into the equation to see if the levels passed fall on or below the trend line (as the trend line indicates a maximum expected number of levels passed for a given number of minutes played).

- **2 minutes played and 3 levels passed**:
\[
y = 0.5(2) - 1 = 1 - 1 = 0 \quad \text{(This is below the trend line)}
\]

- **17 minutes played and 35 levels passed**:
\[
y = 0.5(17) - 1 = 8.5 - 1 = 7.5 \quad \text{(This is way above the trend line)}
\]

- **20 minutes played and 9 levels passed**:
\[
y = 0.5(20) - 1 = 10 - 1 = 9 \quad \text{(This is close to but slightly above the trend line)}
\]

- **50 minutes played and 19 levels passed**:
\[
y = 0.5(50) - 1 = 25 - 1 = 24 \quad \text{(This is well above the trend line)}
\]

From the analyses, the combination that is most likely to be possible based on the trend line is **20 minutes played and 9 levels passed**, as it is the closest to the predicted output from the trend line.