Question

this question has two parts. first, answer part a. then, answer part b.
part a
structure for his project, darius measured the effect of fertilizer on plant growth. at the end of 3 weeks, he recorded the heights of each plant and how many drops of

drops of fertilizer (x)	5	15	20	25	18	22	21	30	10	13	15
height in inches (y)	5	8	9	0	8	0	9	0	7	6	9

a. find the equation of the best - fit line for the data. round to the nearest hundredth, if necessary.
$y = \square x + \square$
part b
b. graph and analyze the residual plot.
the residuals \\(\boldsymbol{\text{select choice}}\\). so, the best - fit line \\(\boldsymbol{\text{select choice}}\\) the data well.
need help?

Explanation:

Step1: Calculate $\bar{x}$ and $\bar{y}$

First, find the mean of $x$ (fertilizer drops) and $y$ (plant height):
$n=11$
$\bar{x} = \frac{5+15+20+25+18+22+21+30+10+13+16}{11} = \frac{195}{11} \approx 17.73$
$\bar{y} = \frac{5+8+9+0+8+0+9+0+7+6+9}{11} = \frac{61}{11} \approx 5.55$

Step2: Calculate slope $m$

Use the formula $m = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$
First compute numerator:
$(5-17.73)(5-5.55)+(15-17.73)(8-5.55)+(20-17.73)(9-5.55)+(25-17.73)(0-5.55)+(18-17.73)(8-5.55)+(22-17.73)(0-5.55)+(21-17.73)(9-5.55)+(30-17.73)(0-5.55)+(10-17.73)(7-5.55)+(13-17.73)(6-5.55)+(16-17.73)(9-5.55)$
$\approx (-12.73)(-0.55)+(-2.73)(2.45)+(2.27)(3.45)+(7.27)(-5.55)+(0.27)(2.45)+(4.27)(-5.55)+(3.27)(3.45)+(12.27)(-5.55)+(-7.73)(1.45)+(-4.73)(0.45)+(-1.73)(3.45)$
$\approx 7.00+(-6.69)+7.83+(-40.35)+0.66+(-23.69)+11.28+(-68.00)+(-11.21)+(-2.13)+(-5.97) \approx -121.27$

Denominator:
$(5-17.73)^2+(15-17.73)^2+(20-17.73)^2+(25-17.73)^2+(18-17.73)^2+(22-17.73)^2+(21-17.73)^2+(30-17.73)^2+(10-17.73)^2+(13-17.73)^2+(16-17.73)^2$
$\approx 162.05+7.45+5.15+52.85+0.07+18.23+10.69+150.55+59.75+22.37+2.99 \approx 492.10$

$m = \frac{-121.27}{492.10} \approx -0.25$

Step3: Calculate y-intercept $b$

Use $b = \bar{y} - m\bar{x}$
$b \approx 5.55 - (-0.25)(17.73) = 5.55 + 4.43 = 9.98$

Step4: Analyze residual plot logic

Residuals are $y - \hat{y}$. For this data, the residuals do not show a random pattern (clusters of positive/negative values with extreme x values having negative residuals), so they are not randomly scattered. This means the line does not fit the data well.

Answer:

Part a:

$y = -0.25x + 9.98$

Part b:

The residuals are not randomly scattered. So, the best-fit line does not model the data well.