Question

a. what is the equation of the line of best fit? round numbers to 2 decimal places.
b. what does the equation estimate for y when x is 2.3? round to 3 decimal places.
c. how does the estimated value compare to the actual value from the table when x is 2.3?
d. how does the estimated value compare to the actual value from the table when x is 3?

Explanation:

Step1: Calculate the means of x and y

Let $x = [2.3,2.8,3.1,3,3.5,3.8]$ and $y=[6.2,5.7,4.7,3.2,3,2.8]$.
The mean of $x$, $\bar{x}=\frac{2.3 + 2.8+3.1+3+3.5+3.8}{6}=\frac{18.5}{6}\approx3.08$
The mean of $y$, $\bar{y}=\frac{6.2 + 5.7+4.7+3.2+3+2.8}{6}=\frac{25.6}{6}\approx4.27$

Step2: Calculate the slope (m)

\[

$$\begin{align*} m&=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\\ \sum_{i = 1}^{6}(x_i-\bar{x})(y_i - \bar{y})&=(2.3 - 3.08)(6.2-4.27)+(2.8 - 3.08)(5.7 - 4.27)+(3.1-3.08)(4.7 - 4.27)+(3 - 3.08)(3.2 - 4.27)+(3.5 - 3.08)(3 - 4.27)+(3.8 - 3.08)(2.8 - 4.27)\\ &=(- 0.78)\times1.93+(-0.28)\times1.43 + 0.02\times0.43+(-0.08)\times(-1.07)+0.42\times(-1.27)+0.72\times(-1.47)\\ &=-1.5054-0.4004 + 0.0086+0.0856-0.5334-1.0584\\ &=-3.4034 \end{align*}$$

\]
\[

$$\begin{align*} \sum_{i=1}^{6}(x_i-\bar{x})^2&=(2.3 - 3.08)^2+(2.8 - 3.08)^2+(3.1-3.08)^2+(3 - 3.08)^2+(3.5 - 3.08)^2+(3.8 - 3.08)^2\\ &=(-0.78)^2+(-0.28)^2+(0.02)^2+(-0.08)^2+(0.42)^2+(0.72)^2\\ &=0.6084 + 0.0784+0.0004+0.0064+0.1764+0.5184\\ &=1.3884 \end{align*}$$

\]
$m=\frac{-3.4034}{1.3884}\approx - 2.45$

Step3: Calculate the y - intercept (b)

$b=\bar{y}-m\bar{x}=4.27-(-2.45)\times3.08=4.27 + 7.546=11.816\approx11.82$
The equation of the line of best - fit is $y=-2.45x + 11.82$

Step4: Estimate y when x = 2.3

Substitute $x = 2.3$ into $y=-2.45x + 11.82$
$y=-2.45\times2.3+11.82=-5.635 + 11.82 = 6.185$

Step5: Compare when x = 2.3

The actual value of y when $x = 2.3$ from the table is $6.2$. The estimated value is $6.185$. The estimated value is less than the actual value by $6.2-6.185 = 0.015$

Step6: Compare when x = 3

Substitute $x = 3$ into $y=-2.45x + 11.82$
$y=-2.45\times3+11.82=-7.35+11.82 = 4.47$
The actual value of y when $x = 3$ from the table is $3.2$. The estimated value is greater than the actual value by $4.47-3.2=1.27$

Answer:

a. $y=-2.45x + 11.82$
b. $6.185$
c. The estimated value ($6.185$) is less than the actual value ($6.2$) by $0.015$
d. The estimated value ($4.47$) is greater than the actual value ($3.2$) by $1.27$