Answer
a) $Y=0.178x+3.17$
b) $5.84$ trillion dollars
c) year $2016$
d) $r=0.998$ indicating a strong positive correlation between $x$ and $y$.
Work Step by Step
a) The least squares lines is given by the equation $Y=mx+b$ where,
$Y:$ The total value of consumer durable goods (in trillions of dollars)
$x:$ No of years since the year $2000$
$m=\frac{n(\sum{xy})-(\sum{x})(\sum{y})}{n\sum{x^2}-(\sum{x})^2}$
$b=\frac{\sum{y}-m\sum{x}}{n}$
Given values:
$n=7$
$\sum{x}=35$
$\sum{x^2}=203$
$\sum{y}=28.4269$
$\sum{y^2}=116.3396$
$\sum{xy}=147.1399$
$m=\frac{(7\times147.1399)-(35\times28.4269)}{(7\times203)-35^2}=0.178$
$b=\frac{28.4269-(0.178\times35)}{7}=3.170$
Therefore, the least squares line can be written as,
$Y=0.178x+3.17$
b) To predict the value of total consumer goods in the year $2015$ using the least squares line obtained in part a).
Since $x$ represents the year we substitute $x=15$ in the least squares equation.
$Y=(0.178\times 15)+3.17=5.84$
The total value of consumer durable goods in the year $2015$ is $5.84$ trillion dollars.
c) To find the year in which the value of consumer durable goods reach $6$ trillion dollars, i.e. $Y=6$.
$Y=0.178x+3.17$
$6=(0.178\times x)+3.17$
$\implies x=15.898 \sim 16$
Therefore, $16$ years since the year $2000$, in the year $2016$, the total value of the goods reaches $6$ trillion dollars.
d) The correlation coefficient,
$r=\frac{n(\sum{xy})-\sum{x}\sum{y}}{\sqrt{n(\sum{x^2})-(\sum{x})^2}\sqrt{n(\sum{y^2})-(\sum{y})^2}}$
$=\frac{(7\times 147.1399)-(35\times 28.4269)}{\sqrt{(7\times 203)-35^2}\sqrt{(7\times 116.3396)-28.4269^2}}=0.998$
$r=0.998$ shows a strong positive linear correlation between the two variables $x$ (No of years since $2000$) and $y$ (total value of consumer durable goods) on a scatterplot. The variables tend to move in the same direction, i.e. as the years move on, the value of the goods also keep on increasing.
