Question

daily sunlight and plant growth
a biologist measures the average hours of sunlight a plant
receives and how many centimeters it grew in a month.
hours of sunlight (x) growth (cm) (y)
3 5
4 8
4 10
5 12
6 14
6 18
7 20
8 22
fill in the blank 1 point
if a plant gets an average of 5.5 hours of sunlight, how much would you
predict it grows?
f(5.5) = type your answer...

Explanation:

Step1: Calculate the mean of \(x\) and \(y\)

\(\bar{x}=\frac{3 + 4+4 + 5+6+6+7+8}{8}=\frac{43}{8} = 5.375\)
\(\bar{y}=\frac{5+8 + 10+12+14+18+20+22}{8}=\frac{109}{8}=13.625\)

Step2: Calculate \(S_{xx}\) and \(S_{xy}\)

\(S_{xx}=\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(3 - 5.375)^{2}+(4 - 5.375)^{2}+(4 - 5.375)^{2}+(5 - 5.375)^{2}+(6 - 5.375)^{2}+(6 - 5.375)^{2}+(7 - 5.375)^{2}+(8 - 5.375)^{2}\)
\(=(- 2.375)^{2}+(-1.375)^{2}+(-1.375)^{2}+(-0.375)^{2}+(0.625)^{2}+(0.625)^{2}+(1.625)^{2}+(2.625)^{2}\)
\(=5.640625+1.890625+1.890625 + 0.140625+0.390625+0.390625+2.640625+6.890625=19.875\)

\(S_{xy}=\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\)
\(=(3 - 5.375)(5 - 13.625)+(4 - 5.375)(8 - 13.625)+(4 - 5.375)(10 - 13.625)+(5 - 5.375)(12 - 13.625)+(6 - 5.375)(14 - 13.625)+(6 - 5.375)(18 - 13.625)+(7 - 5.375)(20 - 13.625)+(8 - 5.375)(22 - 13.625)\)
\(=(-2.375)(-8.625)+(-1.375)(-5.625)+(-1.375)(-3.625)+(-0.375)(-1.625)+(0.625)(0.375)+(0.625)(4.375)+(1.625)(6.375)+(2.625)(8.375)\)
\(=20.53125+7.734375+4.96875+0.609375+0.234375+2.734375+10.390625+21.953125 = 69.15\)

Step3: Calculate the slope \(b\) and intercept \(a\) of the regression line \(y=a + bx\)

\(b=\frac{S_{xy}}{S_{xx}}=\frac{69.15}{19.875}=3.48\)
\(a=\bar{y}-b\bar{x}=13.625-3.48\times5.375=13.625 - 18.78= - 5.155\)

The regression equation is \(y=-5.155 + 3.48x\)

Step4: Predict when \(x = 5.5\)

Substitute \(x = 5.5\) into the regression equation \(y=-5.155+3.48\times5.5\)
\(y=-5.155 + 19.14=13.985\approx14\)

Answer:

\(14\)