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	45.7

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

Explanation:

Part (a)

Step 1: Define Variables

Let \( x \) be the number of years since 2010. So for 2011, \( x = 1 \); 2012, \( x = 2 \); 2013, \( x = 3 \); 2014, \( x = 4 \); 2015, \( x = 5 \). The sales \( y \) (in millions) are: \( (1, 12.2) \), \( (2, 19.1) \), \( (3, 29.4) \), \( (4, 37.3) \), \( (5, 45.7) \).

Step 2: Calculate Mean of \( x \) and \( y \)

Mean of \( x \): \( \bar{x}=\frac{1 + 2 + 3 + 4 + 5}{5}=\frac{15}{5}=3 \)
Mean of \( y \): \( \bar{y}=\frac{12.2 + 19.1 + 29.4 + 37.3 + 45.7}{5}=\frac{143.7}{5}=28.74 \)

Step 3: Calculate Slope (\( m \))

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Step 4: Calculate Intercept (\( b \))

Using \( y = mx + b \) and \( \bar{y}=m\bar{x}+b \)
\( 28.74 = 8.52\times3 + b \)
\( 28.74 = 25.56 + b \)
\( b = 28.74 - 25.56 = 3.18 \)

Step 5: Equation of Best - Fit Line

The equation is \( y = 8.52x + 3.18 \)

Part (b)

Step 1: Formula for Correlation Coefficient (\( r \))

$$ r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}} $$

First, calculate \( \sum x = 15 \), \( \sum y = 143.7 \), \( \sum xy=(1\times12.2)+(2\times19.1)+(3\times29.4)+(4\times37.3)+(5\times45.7)=12.2 + 38.2+88.2 + 149.2+228.5 = 516.3 \)
\( \sum x^2=1^2 + 2^2+3^2 + 4^2+5^2 = 1 + 4 + 9 + 16 + 25 = 55 \)
\( \sum y^2=12.2^2+19.1^2 + 29.4^2+37.3^2+45.7^2=148.84+364.81 + 864.36+1391.29+2088.49 = 4857.79 \)
\( n = 5 \)

Step 2: Calculate Numerator and Denominator

Numerator: \( n\sum xy-\sum x\sum y=5\times516.3-15\times143.7 = 2581.5 - 2155.5 = 426 \)
Denominator:
First, \( n\sum x^2-(\sum x)^2=5\times55 - 15^2=275 - 225 = 50 \)
Second, \( n\sum y^2-(\sum y)^2=5\times4857.79-143.7^2=24288.95 - 20659.69 = 3629.26 \)
\( \sqrt{50\times3629.26}=\sqrt{181463}\approx426 \)

Step 3: Calculate \( r \)

\( r=\frac{426}{426}=1 \) (approximate due to rounding in intermediate steps, more accurately, it is very close to 1)

Step 4: Interpretation

A correlation coefficient \( r\approx1 \) (in this case, very close to 1) indicates a strong positive linear relationship between the number of years since 2010 (\( x \)) and the sales of the company (\( y \)). This means that as the number of years since 2010 increases, the sales of the health and beauty supply company tend to increase in a nearly perfect linear fashion.

Part (a) Answer: \( y = 8.52x + 3.18 \)

Part (b) Answer: The correlation coefficient \( r\approx1 \), indicating a strong positive linear relationship between the number of years since 2010 and the company's sales.

Answer:

Step 1: Formula for Correlation Coefficient (\( r \))

$$ r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}} $$

First, calculate \( \sum x = 15 \), \( \sum y = 143.7 \), \( \sum xy=(1\times12.2)+(2\times19.1)+(3\times29.4)+(4\times37.3)+(5\times45.7)=12.2 + 38.2+88.2 + 149.2+228.5 = 516.3 \)
\( \sum x^2=1^2 + 2^2+3^2 + 4^2+5^2 = 1 + 4 + 9 + 16 + 25 = 55 \)
\( \sum y^2=12.2^2+19.1^2 + 29.4^2+37.3^2+45.7^2=148.84+364.81 + 864.36+1391.29+2088.49 = 4857.79 \)
\( n = 5 \)

Step 2: Calculate Numerator and Denominator

Numerator: \( n\sum xy-\sum x\sum y=5\times516.3-15\times143.7 = 2581.5 - 2155.5 = 426 \)
Denominator:
First, \( n\sum x^2-(\sum x)^2=5\times55 - 15^2=275 - 225 = 50 \)
Second, \( n\sum y^2-(\sum y)^2=5\times4857.79-143.7^2=24288.95 - 20659.69 = 3629.26 \)
\( \sqrt{50\times3629.26}=\sqrt{181463}\approx426 \)

Step 3: Calculate \( r \)

\( r=\frac{426}{426}=1 \) (approximate due to rounding in intermediate steps, more accurately, it is very close to 1)

Step 4: Interpretation

A correlation coefficient \( r\approx1 \) (in this case, very close to 1) indicates a strong positive linear relationship between the number of years since 2010 (\( x \)) and the sales of the company (\( y \)). This means that as the number of years since 2010 increases, the sales of the health and beauty supply company tend to increase in a nearly perfect linear fashion.

Part (a) Answer: \( y = 8.52x + 3.18 \)

Part (b) Answer: The correlation coefficient \( r\approx1 \), indicating a strong positive linear relationship between the number of years since 2010 and the company's sales.