The following scatterplot, with an exponential regression, models a city's population since 1900.   

This scatterplot, with an exponential regression, models a city's population since 1900. The population in 1950 was adjusted because a drought caused an exodus of farmers, most of whom moved back a few years later.
[The graph plots Number of Years since 1900 on the horizontal axis and Population in Thousands on the vertical axis. A curve rises with increasing steepness from the second quadrant, then passes through the points (0, 2.5), (65, 26) in the first quadrant. A series of data points closely follows the curve and the equation near the curve reads: f of x equals 2.38 times e to the power of left parenthesis 0.04 x right parenthesis and r squared equals 0.85.]© 2018 WGU, Powered by GeoGebra 

Question 2 of 12

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Which statement is true about this regression?


Although  f(100)≈124
 , the result should not be trusted.  

By  f(100)≈124
 , it can be estimated that the city had approximately 124,000 residents in 2000. This is an interpolation.  

There are fewer than 30 data points, so no trustworthy conclusion can be made.   

By  f(100)≈124
 , it can be estimated that the city had approximately 124,000 residents in 2000. This is an extrapolation.  

The key aspect of this question is recognizing the difference between **interpolation** and **extrapolation** when making predictions based on a regression model.

### Explanation:
1. The **x-axis** represents the **years since 1900**, so:
   - \(x = 0\) corresponds to **1900**.
   - \(x = 100\) corresponds to **2000**.

2. The **equation** given is \(f(x) = 2.38 \cdot e^{0.04x}\), which models the population in thousands.
   - When \(x = 100\), the function gives \(f(100) \approx 124\), meaning the population in **2000** is estimated to be approximately **124,000** people.

3. **Interpolation vs. Extrapolation**:
   - **Interpolation** involves predicting values within the range of observed data.
   - **Extrapolation** involves predicting values **outside the observed data range**.

   Since the plot involves data primarily covering earlier years (up to **1950** as stated), making a prediction for **2000** (i.e., \(x = 100\)) is an **extrapolation**, as it is beyond the main range of available data points.

### Correct Answer:
- **"By \(f(100) \approx 124\), it can be estimated that the city had approximately 124,000 residents in 2000. This is an extrapolation."**

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Would you like further clarification or additional details?  

Here are five related questions:  
1. What is the difference between linear and exponential regression?  
2. Why is extrapolation generally less reliable than interpolation?  
3. How is the value of \(r^2 = 0.85\) interpreted in this context?  
4. What impact do outliers like the drought (1950) have on regression models?  
5. How can regression models be improved to make more accurate predictions?

**Tip:** In general, the reliability of a prediction decreases the farther you extrapolate beyond the observed data range.

This scatterplot, with an exponential regression, models a city's population since 1900. The population in 1950 was adjusted because a drought caused an exodus of farmers, most of whom moved back a few years later. Which statement is true about this regression?

Explore how exponential regression models population growth over time, with a focus on interpolation and extrapolation. This example looks at a city’s population data from 1900, including how a drought in 1950 affected predictions. Learn how to interpret regression equations, such as f(x) = 2.38 * e^(0.04x), and understand the difference between interpolation and extrapolation.

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