Question

Data were collected on the distance a baseball will travel when hit by a baseball bat at a certain speed. The speed, s, is measured in miles per hour, and distance, y, is measured in yards. The regression line is given by ŷ = 4.17 + 52.45s. Identify the slope and y-intercept of the regression line. Interpret each value in context. Options: Option 1: The slope, 52.45, indicates that the distance increases by 52.45 yards for every one mile per hour of speed. The y-intercept, 4.17, is the distance estimated by this model if the speed is zero miles per hour. Option 2: The slope, 4.17, indicates that the distance increases by 4.17 yards for every one mile per hour of speed. The y-intercept, 52.45, is the distance estimated by this model if the speed is zero miles per hour. Option 3: The slope, 4.17, indicates that the distance decreases by 4.17 yards for every one mile per hour of speed. The y-intercept, 52.45, is the distance estimated by this model if the speed is one mile per hour. Option 4: The slope, 52.45, indicates that the distance decreases by 52.45 yards for every one mile per hour of speed. The y-intercept, 4.17, is the distance estimated by this model if the speed is one mile per hour.

Answer 1

The correct interpretation of the regression line î = 4.17 + 52.45s is that the slope is 52.45 and the y-intercept is 4.17, which means the distance increases by 52.45 yards for each additional mile per hour of speed, and the estimated distance is 4.17 yards when speed is zero.

Step-by-step explanation:

The regression line given by the equation î = 4.17 + 52.45s represents a relationship between the speed of a baseball when hit by a bat (s) and the distance it travels (y). The slope of the regression line is 52.45, which indicates that for every one unit (mile per hour) increase in the speed, the distance the baseball will travel increases by 52.45 yards on average. The y-intercept is 4.17, which represents the distance that the baseball would theoretically travel if the speed was zero miles per hour. It's worth noting that a y-intercept in this context does not always have a logical real-world interpretation, because you can't hit a baseball at zero miles per hour and expect it to travel any distance. However, it is a part of the linear regression model equation.

Considering the options provided, Option 1 is the correct interpretation of the slope and y-intercept in context. Precisely, the slope is 52.45, reflecting the rate of increase in distance as speed increases, and the y-intercept is 4.17, signifying the estimated distance at zero speed.

Answer 2

Option 1 is the correct interpretation.

- The slope (52.45) indicates that for every one mile per hour increase in speed, the distance traveled by the baseball increases by 52.45 yards. This makes sense, as higher speed leads to a longer ball travel distance.

- The y-intercept (4.17) is the estimated distance when the speed is zero miles per hour. While this might not have a direct real-world interpretation (since a baseball doesn't travel when the bat isn't swung), it's still a statistical parameter of the model.

To identify the slope and y-intercept of the regression line ŷ = 4.17 + 52.45s and interpret each value in context, follow these steps:

1. Understand the Regression Line:

- The regression line is in the form of ŷ = mx + b, where ŷ is the predicted value, m is the slope, x is the independent variable (speed in miles per hour), and b is the y-intercept.

2. Identify the Values:

- In the equation ŷ = 4.17 + 52.45s, the values are:

- Slope (m) = 52.45

- Y-intercept (b) = 4.17

3. Interpretation:

- Now, let's interpret each value in context:

Option 1: The slope, 52.45, indicates that the distance increases by 52.45 yards for every one mile per hour of speed. The y-intercept, 4.17, is the distance estimated by this model if the speed is zero miles per hour.

Option 2: The slope, 4.17, indicates that the distance increases by 4.17 yards for every one mile per hour of speed. The y-intercept, 52.45, is the distance estimated by this model if the speed is zero miles per hour.

Option 3: The slope, 4.17, indicates that the distance decreases by 4.17 yards for every one mile per hour of speed. The y-intercept, 52.45, is the distance estimated by this model if the speed is one mile per hour.

Option 4: The slope, 52.45, indicates that the distance decreases by 52.45 yards for every one mile per hour of speed. The y-intercept, 4.17, is the distance estimated by this model if the speed is one mile per hour.