Lesson 5

Fitting Lines

  • Let’s find the best linear model for some data.

Problem 1

Technology required.

\(x\) \(y\)
83 102
87 115
91 107
93 122
97 125
97 127
101 120
104 127
  1. Use graphing technology to create a scatter plot and find the best fit line.
  2. What does the best fit line estimate for the \(y\) value when \(x\) is 100?

Problem 2

Technology required.

\(x\) \(y\)
2.3 6.2
2.8 5.7
3.1 4.7
3 3.2
3.5 3
3.8 2.8
  1. What is the equation of the line of best fit? Round numbers to 2 decimal places.
  2. What does the equation estimate for \(y\) when \(x\) is 2.3? Round to 3 decimal places.
  3. How does the estimated value compare to the actual value from the table when \(x\) is 2.3?
  4. How does the estimated value compare to the actual value from the table when \(x\) is 3?

Problem 3

Which of these scatter plots are best fit by the shown linear model?

A:
A scatter plot. Horizontal from 1 to 7 by 1’s. Vertical from negative 3 to 11 by 1’s. 7 dots with a line of best fit. Trend is up and to the right. Dots are somewhat spaced away from line of best fit.
B:
A scatter plot. Horizontal from 1 to 7 by 1’s. Vertical from negative 3 to 11 by 1’s. 7 dots with a line of best fit. Trend is up and to the right. Dots touching or nearly touching line of best fit.
C:
A scatter plot. Horizontal from 1 to 7 by 1’s. Vertical from negative 3 to 11 by 1’s. 7 dots with a line of best fit. Trend is up and to the right. 1 dot on line of best fit, 3 dots below, 3 dots above.  
D:
A scatter plot. Horizontal from 1 to 7 by 1’s. Vertical from negative 3 to 11 by 1’s. 7 dots with a line of best fit. Trend is down and to the right. 1 dot touching line of best fit; 4 dots below, 2 dots above.  

Problem 4

A seed is planted in a glass pot and its height is measured in centimeters every day.

A scatter plot. Horizontal, 0 to 30, by 1’s, labeled days. Vertical, negative 5 to 3, by point 5’s, labeled height in centimeters. 21 dots trend linearly up and to the right with a line a best fit.

The best fit line is given by the equation \(y = 0.404x-5.18\), where \(y\) represents the height of the plant above ground level, and \(x\) represents the number of days since it was planted.

  1. What is the slope of the best fit line? What does the slope of the line mean in this situation? Is it reasonable?
  2. What is the \(y\)-intercept of the best fit line? What does the \(y\)-intercept of the line mean in this situation? Is it reasonable?
(From Unit 3, Lesson 4.)

Problem 5

At a restaurant, the total bill and the percentage of the bill left as a tip are represented in the scatter plot.

Scatter plot. Horizontal, 0 to 32, by 2’s, labeled total. Vertical, 0 to 32, by 2’s, labeled percentage. 13 dots and a line of best fit trend linearly down and to the right.

The best fit line is represented by the equation \(y = \text{-}0.632x + 27.1\), where \(x\) represents the total bill in dollars, and \(y\) represents the percentage of the bill left as a tip.

  1. What does the best fit line estimate for the percentage of the bill left as a tip when the bill is \$15? Is this reasonable?
  2. What does the best fit line predict for the percentage of the bill left as a tip when the bill is \$50? Is this reasonable?
(From Unit 3, Lesson 4.)

Problem 6

A recent study investigated the amount of battery life remaining in alkaline batteries of different ages. The scatter-plot shows this relationship between the different alkaline batteries tested. 

A scatterplot.

The scatter plot includes a point at \((7, 15)\). Describe the meaning of this point in this situation.

(From Unit 3, Lesson 4.)