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It looks like you’ve uploaded an image with some mathematical problems related to regression analysis. Let me help you with a detailed interpretation and solution for each part of the question. Here’s a summary of what's provided in the image:

### Provided Data Table:
The table shows the time (in hours) and the concentration (in mg/L) of medication in a patient's blood:

| Time (Hours) | Concentration (mg/L) |
|--------------|-----------------------|
| 0            | 0                     |
| 0.5          | 78.1                  |
| 1            | 99.8                  |
| 1.5          | 84.4                  |
| 2            | 50.1                  |
| 2.5          | 15.6                  |

The parts of the question are:

1. **Part a**: Predicting whether a linear or quadratic model would best fit the data based on a scatter plot.
   - The answer is predicted as quadratic.

2. **Part b**: Finding the linear regression equation using a graphing calculator, Excel, or Desmos, and determining the correlation coefficient \( r \) and the coefficient of determination \( r^2 \), rounded to three decimal places.
   - Linear regression equation: \( y = -1.223x + 60.195 \)
   - \( r = -0.029 \) 
   - \( r^2 = 0.172 \)

3. **Part c**: Finding the quadratic regression equation and the coefficient of determination \( r^2 \), rounding to three decimal places.
   - Quadratic regression equation: \( y = -56.714x^2 + 139.313x + 9.35 \)
   - \( r^2 = 0.929 \)

4. **Part d**: Comparing the \( r^2 \) values for linear and quadratic regression to determine which model best fits the data.
   - Conclusion: The quadratic regression fits better because \( r^2 \) for the quadratic model is larger.

5. **Part e**: Using the best-fit regression model to predict the concentration of medication at 1.25 hours and rounding the result to 1 decimal place.
   - Based on the linear regression model: \( y = -1.223(1.25) + 60.195 \)

Let me now calculate the prediction for part **e** using the linear equation.

I'll work that out for you!The predicted concentration of medication in the patient's blood after 1.25 hours, using the linear regression model, is **58.7 mg/L** (rounded to one decimal place).

Would you like any further details or have any questions? Here are 5 additional questions related to this problem:

1. How do you calculate the slope and intercept of a linear regression equation?
2. What is the significance of the \( r^2 \) value in regression analysis?
3. How can you interpret the difference in \( r^2 \) values between linear and quadratic models?
4. Why is it important to round values in regression models?
5. What other types of regression models can be used besides linear and quadratic?

**Tip**: When choosing between regression models, always compare the \( r^2 \) value. A higher \( r^2 \) means a better fit to the data.

Find the best-fit regression model (linear or quadratic) for data related to medication concentration over time, and predict the concentration at 1.25 hours using the best-fit model.

Explore how to determine whether a linear or quadratic regression model best fits the data related to medication concentration over time. Includes step-by-step calculations for linear and quadratic regression equations, correlation coefficient (r), and r^2 values. Suitable for advanced high school and early college students.

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