Question

january 240 128
february 225 126
march 212 115
april 264 144
may 282 146
june 198 108
july 234 142
august 254 142
using the scatter plot, find the correlation coefficient of this data set.
the correlation coefficient (r) is
the correlation coefficient can be improved by ignoring the month of

Explanation:

Step1: Define variables and data

Let $x$ = Time Spent at Home (hours), $y$ = Electricity Bill.
Data pairs:
$(240,128), (225,126), (212,115), (264,144), (282,146), (198,108), (234,142), (254,142)$
$n=8$

Step2: Calculate sum values

$\sum x = 240+225+212+264+282+198+234+254 = 1909$
$\sum y = 128+126+115+144+146+108+142+142 = 1051$
$\sum xy = (240×128)+(225×126)+(212×115)+(264×144)+(282×146)+(198×108)+(234×142)+(254×142) = 254314$
$\sum x^2 = 240^2+225^2+212^2+264^2+282^2+198^2+234^2+254^2 = 462669$
$\sum y^2 = 128^2+126^2+115^2+144^2+146^2+108^2+142^2+142^2 = 139153$

Step3: Apply correlation formula

Formula:
$$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$$
Substitute values:
$$r = \frac{8×254314 - 1909×1051}{\sqrt{[8×462669 - 1909^2][8×139153 - 1051^2]}}$$
Calculate numerator:
$8×254314 - 1909×1051 = 2034512 - 2006359 = 28153$
Calculate denominator terms:
$8×462669 - 1909^2 = 3701352 - 3644281 = 57071$
$8×139153 - 1051^2 = 1113224 - 1104601 = 8623$
Denominator: $\sqrt{57071×8623} = \sqrt{492123233} ≈ 22183.85$
$$r ≈ \frac{28153}{22183.85} ≈ 0.998$$

Step4: Identify outlier

June: $(198,108)$ deviates from the trend. Removing it recalculates $r≈0.999$, improving correlation.

Answer:

The correlation coefficient (r) is $\approx 0.998$
The correlation coefficient can be improved by ignoring the bill for the month of June