Question

1. karl recorded data to find the number of minutes he needs to drive a given number of miles.

part a
find the correlation coefficient of a linear regression for his data. round to the nearest thousandth.
part b
which part of the linear regression would approximate karls average speed for the six trips?
a ( r ), the correlation coefficient
b ( a ); the slope of the line of best fit.
c ( b ); the ( y )-intercept of the line of best fit.
d the ( x )-intercept of the line of best fit.

Explanation:

Part A

The problem provides the correlation coefficient \( r=-0.8945\). When rounding to the nearest thousandth (three decimal places), we look at the fourth decimal digit. The fourth decimal digit of \(-0.8945\) is \(5\). According to the rounding rule (if the digit is \(5\) or greater, we round up the previous digit), so \(-0.8945\approx - 0.895\).

Part B

Step1: Recall the meaning of each component in linear regression

• The correlation coefficient \(r\) measures the strength and direction of a linear relationship between two variables. It does not represent speed.
• The slope \(a\) of the line of best - fit \(y = a x + b\) (where \(y\) is the dependent variable, \(x\) is the independent variable) has a physical interpretation. If \(y\) is the number of miles (distance) and \(x\) is the number of minutes (time), then the slope \(a=\frac{\Delta y}{\Delta x}\). Since speed \(v=\frac{\text{distance}}{\text{time}}\), the slope of the line of best - fit (where \(y\) is distance and \(x\) is time) represents the average speed.
• The \(y\) - intercept \(b\) is the value of \(y\) when \(x = 0\). In the context of distance - time, when \(x = 0\) (time \(=0\)), \(b\) would be the initial distance, not the speed.
• The \(x\) - intercept is the value of \(x\) when \(y = 0\). In the distance - time context, it is the time when the distance is \(0\), not the speed.

Answer:

Part A

\(-0.895\)

Part B

B. \(a\); the slope of the line of best fit.