Question

use technology to perform a linear regression to determine the equation for the line of best fit for the data. estimate the value of y when x = 19.

x	y
14	39
16	41
18	44
20	48

find the line of best fit for the data.
(type your answer in slope-intercept form. use integers or decimals for any numbers in the equation. round to two decimal places as needed.)

Explanation:

Step1: Calculate mean of x and y

First, find the mean of \( x \) values: \( \bar{x}=\frac{12 + 14+16+18+20}{5}=\frac{80}{5} = 16 \)
Mean of \( y \) values: \( \bar{y}=\frac{35 + 39+41+44+48}{5}=\frac{207}{5}=41.4 \)

Step2: Calculate slope (m)

Slope formula: \( m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2} \)
Calculate \( (x_i-\bar{x})(y_i-\bar{y}) \) for each point:

• For (12,35): \( (12 - 16)(35 - 41.4)=(-4)(-6.4) = 25.6 \)
• For (14,39): \( (14 - 16)(39 - 41.4)=(-2)(-2.4)=4.8 \)
• For (16,41): \( (16 - 16)(41 - 41.4)=(0)(-0.4)=0 \)
• For (18,44): \( (18 - 16)(44 - 41.4)=(2)(2.6) = 5.2 \)
• For (20,48): \( (20 - 16)(48 - 41.4)=(4)(6.6)=26.4 \)

Sum of these: \( 25.6+4.8 + 0+5.2+26.4=62 \)

Calculate \( (x_i-\bar{x})^2 \) for each point:

• For 12: \( (-4)^2 = 16 \)
• For 14: \( (-2)^2=4 \)
• For 16: \( 0^2 = 0 \)
• For 18: \( 2^2=4 \)
• For 20: \( 4^2 = 16 \)

Sum of these: \( 16 + 4+0+4+16 = 40 \)

So, \( m=\frac{62}{40}=1.55 \)

Step3: Calculate y-intercept (b)

Using \( \bar{y}=m\bar{x}+b \), so \( b=\bar{y}-m\bar{x} \)
\( b = 41.4-1.55\times16 \)
\( 1.55\times16 = 24.8 \)
\( b=41.4 - 24.8=16.6 \)

Step4: Equation of line

The slope - intercept form is \( y=mx + b \), so \( y = 1.55x+16.6 \)

Step5: Estimate y when x = 19

Substitute \( x = 19 \) into the equation:
\( y=1.55\times19+16.6 \)
\( 1.55\times19 = 29.45 \)
\( y=29.45 + 16.6=46.05 \)

Answer:

The equation of the line of best fit is \( y = 1.55x+16.60 \), and when \( x = 19 \), \( y\approx46.05 \)