QQuestionInformation Technology
QuestionInformation Technology
# Question 6 of 10
## 2 Points
Which regression equation best fits these data?
\begin{array}{cccc}
x & y \\
- 4 & 8 \\
- 3 & 12 \\
- 2 & 14 \\
- 1 & 16 \\
1 & 15 \\
2 & 12 \\
3 & 9 \\
4 & 5
\end{array}
- A. y = 0.58x^2 + 0.43x + 15.75
- B. y = - 0.58x^2 - 0.43x + 15.75
- C. y = 10.72 \cdot 0.95^2
- D. y = - 0.43x + 11.34
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Answer
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Step 1:: Identify the type of regression equation needed.
In this case, the given data points show a curve, so we need to determine if it is a quadratic or exponential curve.
Step 2:: Calculate the sum of squares of residuals (SSR) for both quadratic and exponential models to compare their fits.
Step 3:: Quadratic Model:
So, the quadratic model is $$y = 0.58x^2 + 0.43x + 15.75$$.
Using the given data points, we can set up a system of linear equations as follows: a(- 4)^2 + b(- 4) + c &= 8 \ a(- 3)^2 + b(- 3) + c &= 12 \ a(- 2)^2 + b(- 2) + c &= 14 \ a(- 1)^2 + b(- 1) + c &= 16 \ a(1)^2 + b(1) + c &= 15 \ a(2)^2 + b(2) + c &= 12 \ a(3)^2 + b(3) + c &= 9 \ a(4)^2 + b(4) + c &= 5 \end{cases}
Step 4:: Exponential Model:
So, the exponential model is $$y = 10.72 \cdot 0.95^x$$.
Using the given data points, we can set up a system of nonlinear equations as follows: a(0.95^{- 4}) &= 8 \ a(0.95^{- 3}) &= 12 \ a(0.95^{- 2}) &= 14 \ a(0.95^{- 1}) &= 16 \ a(0.95^{1}) &= 15 \ a(0.95^{2}) &= 12 \ a(0.95^{3}) &= 9 \ a(0.95^{4}) &= 5 \end{cases}
Step 5:: Calculate the SSR for both models.
SSR\_E = \sum\_{i=1}^{n} (y\_i - (ab^{x\_i}))^2
Quadratic Model: Exponential Model:
Step 6:: Compare the SSR values for both models.
Whichever model has the smaller SSR value fits the data better.
Step 7:: Based on the comparison, choose the best-fitting regression equation.
Final Answer
To determine which one is the best fit, follow steps 3 - 6.
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