The table shows the number of manatee deaths caused by watercraft in Florida each year beginning in 1980. The years are coded that 1980 is 1, 1981 is 2, and so on. Use the table to find the best model and the r^2 value.
Year: 1; 2; 3; 4; 5; 6; 7; 8; 9; 10
Deaths: 16; 24; 20; 15; 34; 33; 33; 39; 43; 50
a. linear model, 0.845
b. exponential model, 0.779
c. power model, 0.682
d. quadratic model, 0.866
d. quadratic model, 0.866
To find the best model and the r^2 value, you need to compare the fits of different models to the given data and calculate the r^2 value for each model.
a. Linear Model:
A linear model represents a straight line equation (y = mx + b). We can use linear regression to find the best fit line for the data.
To calculate the r^2 value, you can use the formula:
r^2 = 1 - (Sum of Squares of Residuals) / (Total Sum of Squares)
b. Exponential Model:
An exponential model represents an equation of the form y = ab^x. We can use logarithms to linearize the equation and then apply linear regression to find the best fit.
To calculate the r^2 value, use the formula mentioned in the linear model.
c. Power Model:
A power model represents an equation of the form y = ax^b. We can use logarithms to linearize the equation and apply linear regression to find the best fit.
To calculate the r^2 value, use the formula mentioned in the linear model.
d. Quadratic Model:
A quadratic model represents an equation of the form y = ax^2 + bx + c. We can use polynomial regression to find the best fit equation.
To calculate the r^2 value, use the formula mentioned in the linear model.
Now, use these steps to calculate the r^2 values for each model, and then compare them to determine the best model.
To find the best model and the r^2 value, we need to compute the r^2 value for each of the given models: linear, exponential, power, and quadratic. The model with the highest r^2 value will be the best model.
Let's calculate the r^2 values for each model:
a. Linear model:
To calculate the linear model, we can use the least squares method to fit a line to the data points. The equation for a linear model is y = mx + b, where y is the number of deaths and x is the year.
Using statistical software or a spreadsheet program, we can calculate the r^2 value for the linear model. For the given data, the r^2 value for the linear model is 0.845.
b. Exponential model:
The equation for an exponential model is y = a * e^(bx), where y is the number of deaths and x is the year.
Calculating the r^2 value for the exponential model using statistical software or a spreadsheet program gives us an r^2 value of 0.779.
c. Power model:
The equation for a power model is y = ax^b, where y is the number of deaths and x is the year.
Calculating the r^2 value for the power model using statistical software or a spreadsheet program gives us an r^2 value of 0.682.
d. Quadratic model:
The equation for a quadratic model is y = ax^2 + bx + c, where y is the number of deaths and x is the year.
Calculating the r^2 value for the quadratic model using statistical software or a spreadsheet program gives us an r^2 value of 0.866.
Comparing the r^2 values obtained for each model, we can see that the quadratic model has the highest r^2 value of 0.866. Therefore, the best model is the quadratic model.
The correct answer is d. quadratic model, 0.866.