Question

find the equation for the least squares regression line of the data described below.
mackenzie is an ornithologist who monitors the wintering activity of birds in a forest clearing. she thinks that the number of different species arriving each season is related to the average winter temperature.
for several winters, mackenzie takes precise measurements to compute the average winter temperature (in celsius), x. she also counts the total number of bird species, y, that appeared in the clearing.

average winter temperature (in celsius)	bird species
17	4
19	7
22	7
24	10

round your answers to the nearest thousandth.
y = \\(\square\\)x + \\(\square\\)

Explanation:

Step1: Calculate sums

$n=5$, $\sum x = 15+17+19+22+24 = 97$, $\sum y =6+4+7+7+10 =34$, $\sum xy = (15*6)+(17*4)+(19*7)+(22*7)+(24*10) = 90 + 68 + 133 + 154 + 240 = 685$, $\sum x^2 = 15^2+17^2+19^2+22^2+24^2 =225+289+361+484+576=1935$

Step2: Compute slope $m$

$$\begin{align*} m &= \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \\ &= \frac{5*685 - 97*34}{5*1935 - 97^2} \\ &= \frac{3425 - 3298}{9675 - 9409} \\ &= \frac{127}{266} \approx 0.477 \end{align*}$$

Step3: Compute intercept $b$

$$\begin{align*} b &= \bar{y} - m\bar{x} \\ &= \frac{34}{5} - 0.477*\frac{97}{5} \\ &= 6.8 - 0.477*19.4 \\ &\approx 6.8 - 9.254 = -2.454 \end{align*}$$

Answer:

$y = 0.477x + (-2.454)$