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.