Question

find the linear regression line for the following table of values. you will need to use a calculator, spreadsheet, or statistical software. enter your answer in the form $y = mx + b$, with $m$ and $b$ both rounded to two decimal places. \

$$\begin{tabular}{|c|c|} \\hline x & y \\\\ \\hline 1 & 7.97 \\\\ \\hline 2 & 7.85 \\\\ \\hline 3 & 11.3 \\\\ \\hline 4 & 10 \\\\ \\hline 5 & 12.58 \\\\ \\hline 6 & 15.41 \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: List the data points

We have the following (x, y) pairs: (1, 7.97), (2, 7.85), (3, 11.3), (4, 10), (5, 12.58), (6, 15.41).

Step2: Calculate necessary sums

Using a calculator or software (e.g., Excel, TI - 84, or statistical software like R), we calculate the following:

• Sum of x values: $\sum x=1 + 2+3 + 4+5 + 6=\frac{6\times(6 + 1)}{2}=21$
• Sum of y values: $\sum y=7.97+7.85 + 11.3+10+12.58+15.41 = 65.11$
• Sum of $x\times y$ values: $1\times7.97+2\times7.85+3\times11.3+4\times10+5\times12.58+6\times15.41$

$=7.97+15.7+33.9+40+62.9+92.46=252.93$

• Sum of $x^{2}$ values: $1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}=1 + 4+9+16+25+36 = 91$

Step3: Calculate the slope (m)

The formula for the slope $m$ of the linear regression line $y=mx + b$ is:
$m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}$
where $n = 6$ (number of data points).
Substitute the values:
$m=\frac{6\times252.93-21\times65.11}{6\times91-(21)^{2}}$
$=\frac{1517.58 - 1367.31}{546-441}$
$=\frac{150.27}{105}\approx1.43$

Step4: Calculate the y - intercept (b)

The formula for the y - intercept $b$ is:
$b=\frac{\sum y - m\sum x}{n}$
Substitute $m\approx1.43$, $\sum y = 65.11$, $\sum x=21$, and $n = 6$:
$b=\frac{65.11-1.43\times21}{6}=\frac{65.11 - 30.03}{6}=\frac{35.08}{6}\approx5.85$

Answer:

$y = 1.43x+5.85$