Question

1. suanna collected information about a group of ponies and horses. she made a table showing the height, measured in hands (hh), and the weight, measured in pounds (lbs), of each pony and horse.

write the linear regression equation for this set of data. round all values to the nearest hundredth.
state the correlation coefficient for the linear regression. round your answer to the nearest hundredth.
explain what the correlation coefficient indicates about the linear fit of the data in the context of the problem.

Explanation:

Step1: Input data into calculator

Using a graphing calculator (e.g., TI - 84), enter the \(x\) - values (11, 12, 13, 14, 16, 17) into list \(L_1\) and the \(y\) - values (264, 638, 700, 850, 1230, 1495) into list \(L_2\).

Step2: Calculate linear regression equation

On the calculator, go to the STAT → CALC menu and select LinReg(ax + b). The calculator will output the values of \(a\) (slope) and \(b\) (y - intercept).

$$
LATEXBLOCK0
$$

The linear regression equation is \(y = 127.65x-1133.43\).

Step3: Calculate correlation coefficient

When running the LinReg(ax + b) command on the calculator, it also outputs the correlation coefficient \(r\).

$$r\approx0.99$$

Step4: Interpret correlation coefficient

The correlation coefficient \(r\) measures the strength and direction of a linear relationship between two variables. The value of \(r\) ranges from \(- 1\) to \(1\). A value of \(r = 1\) indicates a perfect positive linear relationship, \(r=-1\) indicates a perfect negative linear relationship, and \(r = 0\) indicates no linear relationship.

Since \(r\approx0.99\) (close to \(1\)), there is a very strong positive linear relationship between the height (\(x\)) and weight (\(y\)) of the ponies and horses. As the height (in hands) increases, the weight (in pounds) also tends to increase in a nearly linear fashion.

Answer:

• Linear regression equation: \(y = 127.65x - 1133.43\)
• Correlation coefficient: \(r\approx0.99\)
• Explanation: A correlation coefficient of approximately \(0.99\) (close to \(1\)) indicates a very strong positive linear relationship. So, as the height of ponies and horses (in hands) increases, their weight (in pounds) has a strong tendency to increase in a linear manner.