Question

1. sales the table shows the sales of a health and beauty supply company in millions of dollars, for several years. let x be the number of years since 2010.

year	2011	2012	2013	2014	2015
sales	12.2	19.1	29.4	37.3	49.7

a. write the equation for the best fit line for the data
b. find and interpret the correlation coefficient

Explanation:

Step1: Define Variables

Let \( x \) be years since 2010, so for 2011, \( x = 1 \); 2012, \( x = 2 \); 2013, \( x = 3 \); 2014, \( x = 4 \); 2015, \( x = 5 \). Sales \( y \) (in millions): \( (1, 12.2), (2, 19.1), (3, 29.4), (4, 37.3), (5, 49.7) \).

Step2: Calculate Mean of \( x \) and \( y \)

\( \bar{x} = \frac{1 + 2 + 3 + 4 + 5}{5} = 3 \)
\( \bar{y} = \frac{12.2 + 19.1 + 29.4 + 37.3 + 49.7}{5} = \frac{147.7}{5} = 29.54 \)

Step3: Calculate Slope (\( m \)) of Best - Fit Line

\( m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2} \)
First, calculate \( (x_i-\bar{x})(y_i - \bar{y}) \) and \( (x_i-\bar{x})^2 \) for each \( i \):

• For \( i = 1 \): \( (1 - 3)(12.2 - 29.54)=(-2)(-17.34) = 34.68 \); \( (1 - 3)^2=4 \)
• For \( i = 2 \): \( (2 - 3)(19.1 - 29.54)=(-1)(-10.44) = 10.44 \); \( (2 - 3)^2 = 1 \)
• For \( i = 3 \): \( (3 - 3)(29.4 - 29.54)=(0)(-0.14)=0 \); \( (3 - 3)^2 = 0 \)
• For \( i = 4 \): \( (4 - 3)(37.3 - 29.54)=(1)(7.76)=7.76 \); \( (4 - 3)^2 = 1 \)
• For \( i = 5 \): \( (5 - 3)(49.7 - 29.54)=(2)(20.16)=40.32 \); \( (5 - 3)^2 = 4 \)

\( \sum(x_i-\bar{x})(y_i - \bar{y})=34.68 + 10.44+0 + 7.76 + 40.32=93.2 \)
\( \sum(x_i-\bar{x})^2=4 + 1+0 + 1+4 = 10 \)
\( m=\frac{93.2}{10}=9.32 \)

Step4: Calculate Intercept (\( b \)) of Best - Fit Line

Using \( y=mx + b \) and \( \bar{y}=m\bar{x}+b \)
\( 29.54=9.32\times3 + b \)
\( 29.54 = 27.96 + b \)
\( b=29.54 - 27.96 = 1.58 \)
So the best - fit line equation is \( y = 9.32x+1.58 \)

Step5: Calculate Correlation Coefficient (\( r \))

\( r=\frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2\sum(y_i - \bar{y})^2}} \)
First, calculate \( \sum(y_i - \bar{y})^2 \):

• For \( i = 1 \): \( (12.2 - 29.54)^2=(-17.34)^2 = 300.6756 \)
• For \( i = 2 \): \( (19.1 - 29.54)^2=(-10.44)^2 = 108.9936 \)
• For \( i = 3 \): \( (29.4 - 29.54)^2=(-0.14)^2 = 0.0196 \)
• For \( i = 4 \): \( (37.3 - 29.54)^2=(7.76)^2 = 60.2176 \)
• For \( i = 5 \): \( (49.7 - 29.54)^2=(20.16)^2 = 406.4256 \)

\( \sum(y_i - \bar{y})^2=300.6756+108.9936 + 0.0196+60.2176+406.4256 = 876.332 \)
\( r=\frac{93.2}{\sqrt{10\times876.332}}=\frac{93.2}{\sqrt{8763.32}}\approx\frac{93.2}{93.61}\approx0.995 \)
Interpretation: \( r\approx0.995 \) is close to 1, indicating a strong positive linear relationship between years since 2010 and sales.

Answer:

a. The equation of the best - fit line is \( \boldsymbol{y = 9.32x + 1.58} \)
b. The correlation coefficient \( r\approx\boldsymbol{0.995} \), which indicates a very strong positive linear association between the number of years since 2010 and the company's sales (as \( r \) is close to 1, the linear relationship between \( x \) and \( y \) is very strong).