Question

question 8
the domain of the data given is the number of rainy days each year in a town from 2010 to 2022. the range of the data given is the number of inches of rain that fell each corresponding year.

x	102	126	115	103	114	107	125	125	118	95	95	59	105
y	32.3	49.5	60.3	55.1	52.5	79.2	56.9	77.1	41.2	43.1	51.9	25.4	47.0

perform a linear regression on the data and write the equation for the line of best fit. round your values to the nearest thousandth.
( y = square x + square )

Explanation:

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

First, find the mean of $x$ values and $y$ values:
$\bar{x} = \frac{102+126+115+103+114+107+125+125+118+95+95+59+105}{13} = \frac{1389}{13} \approx 106.846$
$\bar{y} = \frac{32.3+49.5+60.3+55.1+52.5+79.2+56.9+77.1+41.2+43.1+51.9+25.4+47.0}{13} = \frac{671.5}{13} \approx 51.654$

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:
$\sum(x_i-\bar{x})(y_i-\bar{y}) = (102-106.846)(32.3-51.654)+(126-106.846)(49.5-51.654)+...+(105-106.846)(47.0-51.654) \approx 3924.538$
Denominator:
$\sum(x_i-\bar{x})^2 = (102-106.846)^2+(126-106.846)^2+...+(105-106.846)^2 \approx 9277.692$
$m = \frac{3924.538}{9277.692} \approx 0.423$

Step3: Calculate intercept $b$

Use $b = \bar{y} - m\bar{x}$
$b = 51.654 - (0.423)(106.846) \approx 51.654 - 44.406 \approx 7.252$

Step4: Write regression equation

Combine slope and intercept:
$y = 0.423x + 7.252$

Answer:

$y = 0.423x + 7.252$