Question

1. (6 points) given the following 5 pairs of observations: (0,4), (1,5), (3,8), (4,7), (2,6)

a) calculate the correlation coefficient ( r ) of the data set.
b) determine the equation of the best fitted line for this data set.

Explanation:

Part (a)

Step 1: List the data points

Let the data points be \((x_i, y_i)\) where \(i = 1,2,3,4,5\): \((0,4)\), \((1,5)\), \((3,8)\), \((4,7)\), \((2,6)\)

Step 2: Calculate necessary sums

• \(n = 5\)
• \(\sum x_i=0 + 1+3 + 4+2=10\)
• \(\sum y_i=4 + 5+8 + 7+6=30\)
• \(\sum x_i^2=0^2+1^2 + 3^2+4^2+2^2=0 + 1+9 + 16+4 = 30\)
• \(\sum y_i^2=4^2+5^2 + 8^2+7^2+6^2=16 + 25+64 + 49+36 = 190\)
• \(\sum x_iy_i=(0\times4)+(1\times5)+(3\times8)+(4\times7)+(2\times6)=0 + 5+24 + 28+12 = 69\)

Step 3: Use the correlation coefficient formula

The formula for the correlation coefficient \(r\) is:
\[r=\frac{n\sum x_iy_i-\sum x_i\sum y_i}{\sqrt{[n\sum x_i^2 - (\sum x_i)^2][n\sum y_i^2-(\sum y_i)^2]}}\]

Substitute the values:

• Numerator: \(5\times69-10\times30 = 345 - 300=45\)
• Denominator part 1: \(5\times30-(10)^2=150 - 100 = 50\)
• Denominator part 2: \(5\times190-(30)^2=950 - 900 = 50\)
• Denominator: \(\sqrt{50\times50}=\sqrt{2500} = 50\)

So, \(r=\frac{45}{50}=0.9\)

Step 1: Calculate the slope \(m\) and the y - intercept \(b\) of the best - fit line \(y=mx + b\)

The formula for the slope \(m\) is:
\[m=\frac{n\sum x_iy_i-\sum x_i\sum y_i}{n\sum x_i^2-(\sum x_i)^2}\]
We already know from part (a) that \(n\sum x_iy_i-\sum x_i\sum y_i = 45\) and \(n\sum x_i^2-(\sum x_i)^2=50\)
So, \(m=\frac{45}{50}=0.9\)

The formula for the y - intercept \(b\) is:
\[b=\bar{y}-m\bar{x}\]
Where \(\bar{x}=\frac{\sum x_i}{n}=\frac{10}{5} = 2\) and \(\bar{y}=\frac{\sum y_i}{n}=\frac{30}{5}=6\)

Substitute \(m = 0.9\), \(\bar{x}=2\) and \(\bar{y}=6\) into the formula for \(b\):
\(b=6-0.9\times2=6 - 1.8 = 4.2\)

Step 2: Write the equation of the best - fit line

The equation of the best - fit line is \(y=0.9x + 4.2\)

Answer:

The correlation coefficient \(r = 0.9\)

Part (b)