Question

modeling with mathematics the table shows the total numbers y of people who volunteer at an animal shelter on each day x.
day, x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
people, y | 9 | 5 | 13 | 11 | 10 | 11 | 19 | 12
a. use a graphing calculator to find the equation of the line of best fit. round all values to the nearest integer.
y =
b. identify the correlation coefficient. round to the nearest thousandth.
r =

Explanation:

Step1: Calculate mean of x and y

First, find the mean of \( x \) values: \( x = [1, 2, 3, 4, 5, 6, 7, 8] \), so \( \bar{x}=\frac{1 + 2+3+4+5+6+7+8}{8}=\frac{36}{8} = 4.5 \)
Mean of \( y \) values: \( y = [9, 5, 13, 11, 10, 11, 19, 12] \), so \( \bar{y}=\frac{9 + 5+13+11+10+11+19+12}{8}=\frac{89}{8}=11.125 \)

Step2: Calculate slope (m)

The formula for slope \( m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2} \)
Calculate \( (x_i-\bar{x})(y_i-\bar{y}) \) for each \( i \):

• \( i = 1 \): \( (1 - 4.5)(9 - 11.125)=(- 3.5)(-2.125)=7.4375 \)
• \( i = 2 \): \( (2 - 4.5)(5 - 11.125)=(-2.5)(-6.125)=15.3125 \)
• \( i = 3 \): \( (3 - 4.5)(13 - 11.125)=(-1.5)(1.875)=-2.8125 \)
• \( i = 4 \): \( (4 - 4.5)(11 - 11.125)=(-0.5)(-0.125)=0.0625 \)
• \( i = 5 \): \( (5 - 4.5)(10 - 11.125)=(0.5)(-1.125)=-0.5625 \)
• \( i = 6 \): \( (6 - 4.5)(11 - 11.125)=(1.5)(-0.125)=-0.1875 \)
• \( i = 7 \): \( (7 - 4.5)(19 - 11.125)=(2.5)(7.875)=19.6875 \)
• \( i = 8 \): \( (8 - 4.5)(12 - 11.125)=(3.5)(0.875)=3.0625 \)

Sum of these products: \( 7.4375+15.3125-2.8125 + 0.0625-0.5625-0.1875+19.6875+3.0625=42 \)

Calculate \( (x_i - \bar{x})^2 \) for each \( i \):

• \( i = 1 \): \( (1 - 4.5)^2=12.25 \)
• \( i = 2 \): \( (2 - 4.5)^2 = 6.25 \)
• \( i = 3 \): \( (3 - 4.5)^2=2.25 \)
• \( i = 4 \): \( (4 - 4.5)^2=0.25 \)
• \( i = 5 \): \( (5 - 4.5)^2=0.25 \)
• \( i = 6 \): \( (6 - 4.5)^2=2.25 \)
• \( i = 7 \): \( (7 - 4.5)^2=6.25 \)
• \( i = 8 \): \( (8 - 4.5)^2=12.25 \)

Sum of these squares: \( 12.25+6.25 + 2.25+0.25+0.25+2.25+6.25+12.25 = 42 \)

So \( m=\frac{42}{42}=1 \) (rounded to nearest integer)

Step3: Calculate y-intercept (b)

Using \( \bar{y}=m\bar{x}+b \), so \( b=\bar{y}-m\bar{x} \)
Substitute \( \bar{y}=11.125 \), \( m = 1 \), \( \bar{x}=4.5 \)
\( b=11.125-1\times4.5 = 6.625\approx7 \) (rounded to nearest integer)

So the equation of the line of best fit is \( y=x + 7 \)

Step4: Calculate correlation coefficient (r)

The formula for correlation coefficient \( r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i-\bar{y})^2}} \)
We already know \( \sum(x_i-\bar{x})(y_i-\bar{y}) = 42 \) and \( \sum(x_i-\bar{x})^2=42 \)
Calculate \( \sum(y_i-\bar{y})^2 \):

• \( i = 1 \): \( (9 - 11.125)^2=(-2.125)^2 = 4.515625 \)
• \( i = 2 \): \( (5 - 11.125)^2=(-6.125)^2=37.515625 \)
• \( i = 3 \): \( (13 - 11.125)^2=(1.875)^2 = 3.515625 \)
• \( i = 4 \): \( (11 - 11.125)^2=(-0.125)^2=0.015625 \)
• \( i = 5 \): \( (10 - 11.125)^2=(-1.125)^2 = 1.265625 \)
• \( i = 6 \): \( (11 - 11.125)^2=(-0.125)^2=0.015625 \)
• \( i = 7 \): \( (19 - 11.125)^2=(7.875)^2=62.015625 \)
• \( i = 8 \): \( (12 - 11.125)^2=(0.875)^2 = 0.765625 \)

Sum of these squares: \( 4.515625+37.515625+3.515625+0.015625+1.265625+0.015625+62.015625+0.765625 = 109.625 \)

Then \( r=\frac{42}{\sqrt{42\times109.625}}=\frac{42}{\sqrt{4604.25}}\approx\frac{42}{67.86}\approx0.619 \) (rounded to nearest thousandth)

Answer:

a. \( y = x + 7 \)
b. \( r = 0.619 \)