Question

question 13 (multiple choice worth 1 points)
(linear functions fit to data mc)

the graph shows a scatter plot for a set of data.

which of the following equations is the best model for a line of fit for the data?

\\(\hat{y} = -1.31x + 0.02\\)
\\(\hat{y} = 1.31x + 0.02\\)
\\(\hat{y} = -0.82x + 1.98\\)
\\(\hat{y} = 0.82x + 1.98\\)

Explanation:

Determine the sign of the slope

The scatter plot shows that as the width (\(x\)) increases, the length (\(y\)) also increases. This indicates a positive correlation, meaning the slope (\(m\)) of the line of best fit must be positive:

$$ m > 0 $$

Estimate the slope and intercept using data points

Select two representative points from the trend of the data:

$$ (x_1, y_1) \approx (1.3, 1.8) $$

$$ (x_2, y_2) \approx (2.7, 3.5) $$

Calculate the approximate slope:

$$ m \approx \frac{3.5 - 1.8}{2.7 - 1.3} = \frac{1.7}{1.4} \approx 1.21 $$

Calculate the approximate \(y\)-intercept using \(y - mx = b\):

$$ b \approx 1.8 - 1.21(1.3) = 1.8 - 1.573 = 0.227 $$

Compare with the given options

Compare the calculated values \(m \approx 1.21\) and \(b \approx 0.23\) with the positive-slope options:

• \(\hat{y} = 1.31x + 0.02\) (Slope \(1.31\), Intercept \(0.02\))
• \(\hat{y} = 0.82x + 1.98\) (Slope \(0.82\), Intercept \(1.98\))

The option \(\hat{y} = 1.31x + 0.02\) is the closest match to the data.

Answer:

• (A) \(\hat{y} = -1.31x + 0.02\)
• (B) \(\hat{y} = 1.31x + 0.02\) (Correct answer)
• (C) \(\hat{y} = -0.82x + 1.98\)
• (D) \(\hat{y} = 0.82x + 1.98\)