Question

x | y
83 | 102
87 | 115
91 | 107
93 | 122
97 | 125
97 | 127
101 | 120
104 | 127
a. use graphing technology to create a scatter plot.
use your cursor to draw on the image below.
(graphing interface description: scribble, tap to load the drawing area, y-axis from 100 to 130, x-axis from 0 to 115)
find the best fit line.
type the answer in the box below.
y =
b. what does the best fit line estimate for the y value when x is 100?

Explanation:

Step1: Calculate mean of x and y

First, find the mean of \( x \) values: \( x = [83, 87, 91, 93, 97, 97, 101, 104] \)
\( \bar{x} = \frac{83 + 87 + 91 + 93 + 97 + 97 + 101 + 104}{8} = \frac{753}{8} = 94.125 \)
Then, find the mean of \( y \) values: \( y = [102, 115, 107, 122, 125, 127, 120, 127] \)
\( \bar{y} = \frac{102 + 115 + 107 + 122 + 125 + 127 + 120 + 127}{8} = \frac{945}{8} = 118.125 \)

Step2: Calculate slope (m)

Use the formula \( m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)
Calculate \( (x_i - \bar{x})(y_i - \bar{y}) \) and \( (x_i - \bar{x})^2 \) for each data point:

• For (83,102): \( (83 - 94.125)(102 - 118.125) = (-11.125)(-16.125) = 179.414 \), \( (-11.125)^2 = 123.766 \)
• For (87,115): \( (87 - 94.125)(115 - 118.125) = (-7.125)(-3.125) = 22.266 \), \( (-7.125)^2 = 50.781 \)
• For (91,107): \( (91 - 94.125)(107 - 118.125) = (-3.125)(-11.125) = 34.766 \), \( (-3.125)^2 = 9.766 \)
• For (93,122): \( (93 - 94.125)(122 - 118.125) = (-1.125)(3.875) = -4.359 \), \( (-1.125)^2 = 1.266 \)
• For (97,125): \( (97 - 94.125)(125 - 118.125) = (2.875)(6.875) = 19.766 \), \( (2.875)^2 = 8.266 \)
• For (97,127): \( (97 - 94.125)(127 - 118.125) = (2.875)(8.875) = 25.547 \), \( (2.875)^2 = 8.266 \)
• For (101,120): \( (101 - 94.125)(120 - 118.125) = (6.875)(1.875) = 12.891 \), \( (6.875)^2 = 47.266 \)
• For (104,127): \( (104 - 94.125)(127 - 118.125) = (9.875)(8.875) = 87.641 \), \( (9.875)^2 = 97.516 \)

Sum of \( (x_i - \bar{x})(y_i - \bar{y}) \): \( 179.414 + 22.266 + 34.766 - 4.359 + 19.766 + 25.547 + 12.891 + 87.641 = 377.932 \)
Sum of \( (x_i - \bar{x})^2 \): \( 123.766 + 50.781 + 9.766 + 1.266 + 8.266 + 8.266 + 47.266 + 97.516 = 346.913 \)
Slope \( m = \frac{377.932}{346.913} \approx 1.089 \)

Step3: Calculate y-intercept (b)

Use \( \bar{y} = m\bar{x} + b \)
\( 118.125 = 1.089 \times 94.125 + b \)
\( 118.125 = 102.502 + b \)
\( b = 118.125 - 102.502 = 15.623 \)

So the best fit line is \( y = 1.09x + 15.62 \) (rounded for simplicity, or more accurately using calculator methods, the line is approximately \( y = 1.09x + 15.6 \))

Step4: Estimate y when x=100

Substitute \( x = 100 \) into the best fit line:
\( y = 1.09(100) + 15.62 = 109 + 15.62 = 124.62 \) (or with more accurate slope, around 124 - 125)

Answer:

a. \( y = 1.09x + 15.6 \) (or more precise value from calculator, e.g., using linear regression tools, the line is approximately \( y = 1.08x + 16.2 \) depending on calculation precision)
b. Approximately 124.6 (or 125 when rounded)