Question

which model is the best? linear vs exponential
the data below shows the amount of birds in an area measured over several weeks

weeks passed	0	1	2	3	4	5	6
birds	40	49	62	71	78	88	102

1. find the linear regression equation for the data above
2. find the correlation coefficient for the linear regression model. is it positive/negative? is it strong/weak?
3. according to the linear regression model, how many birds will be present after 10 weeks?
4. find the exponential regression equation for the data above
5. find the correlation coefficient for the exponential regression model. is it positive/negative? is it strong/weak?
6. according to the exponential regression model, how many birds will be present after 10 weeks?
7. is the estimate from question 3 or question 6 more accurate? how do you know?

Explanation:

Step1: Define Variables

Let \( x \) be the weeks passed (\( x = [0, 1, 2, 3, 4, 5, 6] \)) and \( y \) be the number of birds (\( y = [40, 49, 62, 71, 78, 88, 102] \)). We use a calculator or software (e.g., TI - 84, Excel, Python) to perform linear regression. The linear regression equation is of the form \( \hat{y}=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

Using a calculator (for example, in a TI - 84: enter the data into lists, then use LinReg(ax + b) function):

• \( n = 7 \) (number of data points)
• \( \sum x=0 + 1+2 + 3+4 + 5+6=\frac{6\times(6 + 1)}{2}=21 \)
• \( \sum y=40+49 + 62+71+78+88+102 = 490 \)
• \( \sum xy=(0\times40)+(1\times49)+(2\times62)+(3\times71)+(4\times78)+(5\times88)+(6\times102)=0 + 49+124+213+312+440+612 = 1750 \)
• \( \sum x^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}=0 + 1+4+9+16+25+36 = 91 \)

The formula for the slope \( m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}} \) and the y - intercept \( b=\frac{\sum y - m\sum x}{n} \)

Substitute the values:
\( m=\frac{7\times1750-21\times490}{7\times91 - 21^{2}}=\frac{12250-10290}{637 - 441}=\frac{1960}{196}=10 \)
\( b=\frac{490-10\times21}{7}=\frac{490 - 210}{7}=\frac{280}{7}=40 \)

Step2: Write the Equation

The linear regression equation is \( \hat{y}=10x + 40 \)

Answer:

The Linear Regression Equation is \( \hat{y}=10x + 40 \) (where \( x \) is the number of weeks passed and \( \hat{y} \) is the predicted number of birds)