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	45.1

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 the number of years since 2010. So for 2011, \( x = 1 \); 2012, \( x = 2 \); 2013, \( x = 3 \); 2014, \( x = 4 \); 2015, \( x = 5 \). Let \( y \) be the sales in millions of dollars. The data points are \((1, 12.2)\), \((2, 19.1)\), \((3, 29.4)\), \((4, 37.3)\), \((5, 45.1)\).

Step2: Calculate necessary sums

First, calculate \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \).

• \( \sum x = 1 + 2 + 3 + 4 + 5 = 15 \)
• \( \sum y = 12.2 + 19.1 + 29.4 + 37.3 + 45.1 = 143.1 \)
• \( \sum xy = (1)(12.2) + (2)(19.1) + (3)(29.4) + (4)(37.3) + (5)(45.1) = 12.2 + 38.2 + 88.2 + 149.2 + 225.5 = 513.3 \)
• \( \sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55 \)

Step3: Calculate slope (\(m\)) and y-intercept (\(b\)) for best - fit line

The formula for the slope \( m \) of the best - fit line \( y=mx + b \) is \( m=\frac{n\sum xy-\sum x\sum y}{n\sum x^2 - (\sum x)^2} \), and the formula for the y - intercept \( b=\frac{\sum y - m\sum x}{n} \), where \( n = 5 \) (number of data points).

First, calculate \( m \):

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Then, calculate \( b \):

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So the equation of the best - fit line is \( y = 8.4x+3.42 \).

Step4: Calculate the correlation coefficient (\(r\))

The formula for the correlation coefficient \( r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2 - (\sum y)^2]}} \). First, we need to calculate \( \sum y^2 \).

\( \sum y^2=(12.2)^2+(19.1)^2+(29.4)^2+(37.3)^2+(45.1)^2 \)
\(=148.84+364.81 + 864.36+1391.29+2034.01=4803.31 \)

Now, calculate the numerator: \( n\sum xy-\sum x\sum y=5\times513.3 - 15\times143.1=2566.5 - 2146.5 = 420 \)

Calculate the denominator:
First, \( n\sum x^2-(\sum x)^2=5\times55 - 15^2=275 - 225 = 50 \)
Second, \( n\sum y^2-(\sum y)^2=5\times4803.31-143.1^2=24016.55 - 20477.61 = 3538.94 \)
Then, \( \sqrt{50\times3538.94}=\sqrt{176947}\approx420.65 \)

So, \( r=\frac{420}{420.65}\approx0.998 \)

Answer:

a. The equation of the best - fit line is \( y = 8.4x+3.42 \)

b. The correlation coefficient \( r\approx0.998 \). This value is very close to 1, which indicates a very strong positive linear relationship between the number of years since 2010 and the sales of the health and beauty supply company.