What is the quadratic regression equation that fits these data?
| Number of seconds | Height (in feet) |
| ----------------- | ---------------- |
| 0 | 12 |
| 1 | 14 |
| 2 | 15 |
| 3 | 14 |
| 4 | 10 |
| 5 | 6 |
A. [tex]y=-3.69 x^2+2.08 x+8.07[/tex]
B. [tex]y=-1.94 x^2+0.62 x+9.62[/tex]
C. [tex]y=15.68(0.88^x)[/tex]
Answer (2)
Input the given data points into a statistical software or calculator.
Use the quadratic regression function to find the coefficients a, b, and c.
The coefficients are approximately: a ≈ − 1.94 , b ≈ 0.62 , c ≈ 9.62 .
The quadratic regression equation is: y = − 1.94 x 2 + 0.62 x + 9.62 .
Explanation
Understanding the Problem We are given a set of data points representing the height of an object at different times and asked to find the quadratic regression equation that best fits this data. The general form of a quadratic equation is y = a x 2 + b x + c , where a , b , and c are constants. Our goal is to determine the values of these constants that provide the best fit for the given data.
Finding the Regression Equation To find the quadratic regression equation, we can use statistical software or a calculator with statistical functions. Input the data points (0, 12), (1, 14), (2, 15), (3, 14), (4, 10), and (5, 6) into the software or calculator. Then, use the quadratic regression function to calculate the coefficients a , b , and c .
Determining the Coefficients After performing the quadratic regression, we obtain the following coefficients: a \[ a pp ro x ] − 1.94 b \[ a pp ro x ] 0.62 c ≈ 9.62 Therefore, the quadratic regression equation is approximately y = − 1.94 x 2 + 0.62 x + 9.62 .
Selecting the Correct Option Comparing the obtained equation with the given options, we find that it matches option B: y = − 1.94 x 2 + 0.62 x + 9.62
Final Answer The quadratic regression equation that fits the given data is y = − 1.94 x 2 + 0.62 x + 9.62 .
Examples
Quadratic regression is useful in many real-world scenarios. For example, if you are analyzing the trajectory of a ball thrown in the air, you can use quadratic regression to model its path. By collecting data points of the ball's height at different times, you can create a quadratic equation that approximates its trajectory. This equation can then be used to predict the ball's height at any given time, or to determine when it will hit the ground. Similarly, quadratic regression can be used to model other phenomena that exhibit a curved relationship, such as the growth of a population or the relationship between temperature and reaction rate in a chemical process.
The quadratic regression equation that fits the provided data is approximately y = − 1.94 x 2 + 0.62 x + 9.62 , which matches option B. This equation models the height as a function of time and describes how height changes over the provided seconds. By using statistical software to perform the regression analysis, we obtain these coefficients accurately.
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