Question

construct the equation of the regression line.

an editing firm compiled the following table which lists the number of pages contained in a piece of technical writing and the cost of proofreading and correcting them (in dollars). assume there is a significant linear relationship between x and y and construct the equation of the linear regression line.

number of pages vs cost

number of pages, x	7	12	4	14	25	30

note: r=0.9999

• ŷ = 15.4(x) + 7.1
• ŷ = 7.1(x) + 15.4
• ŷ = 1.6(x) + 17.9
• ŷ = 17.9(x) + 1.6

Explanation:

Step1: Calculate mean of x and y

First, find the mean of \( x \) (number of pages) and \( y \) (cost).
For \( x \): \( \bar{x} = \frac{7 + 12 + 4 + 14 + 25 + 30}{6} = \frac{92}{6} \approx 15.333 \)
For \( y \): \( \bar{y} = \frac{128 + 213 + 75 + 250 + 446 + 540}{6} = \frac{1652}{6} \approx 275.333 \)

Step2: Calculate slope (m)

The slope \( m \) of the regression line can also be estimated by checking which option passes near \( (\bar{x}, \bar{y}) \). Let's test each option:

• Option 1: \( \hat{y} = 15.4x + 7.1 \). At \( x \approx 15.333 \), \( \hat{y} = 15.4(15.333) + 7.1 \approx 236.13 + 7.1 \approx 243.23 \) (not close to 275.33)
• Option 2: \( \hat{y} = 7.1x + 15.4 \). At \( x \approx 15.333 \), \( \hat{y} = 7.1(15.333) + 15.4 \approx 108.86 + 15.4 \approx 124.26 \) (no)
• Option 3: \( \hat{y} = 1.6x + 17.9 \). At \( x \approx 15.333 \), \( \hat{y} = 1.6(15.333) + 17.9 \approx 24.53 + 17.9 \approx 42.43 \) (no)
• Option 4: \( \hat{y} = 17.9x + 1.6 \). At \( x \approx 15.333 \), \( \hat{y} = 17.9(15.333) + 1.6 \approx 274.46 + 1.6 \approx 276.06 \), which is very close to \( \bar{y} \approx 275.33 \).

Answer:

\( \hat{y} = 17.9(x) + 1.6 \)