(a) Exponential Regression Model

We are tasked with using exponential regression to model the price $P(t)$ of a two-bedroom apartment in downtown New York City as a function of time $t$ (where $t = 0$ represents 1994).

Given data points:

• $t = 0$ (1994), Price = 0.38 million
• $t = 2$, Price = 0.40 million
• $t = 4$, Price = 0.60 million
• $t = 6$, Price = 0.95 million
• $t = 8$, Price = 1.20 million
• $t = 10$ (2004), Price = 1.60 million

The general form of an exponential model is: $P(t) = a e^{bt}$ Where:

• $a$ is the initial value (price at $t = 0$),
• $b$ is the growth rate,
• $t$ is the time since 1994,
• $P(t)$ is the price of the apartment at time $t$.

Step-by-Step Process for Exponential Regression:

1. Fit the exponential regression curve to the given data points. This requires finding the values of $a$ and $b$ by applying exponential regression techniques (or using a calculator/software).

2. Exponential regression using technology (e.g., graphing calculator, Excel, or any statistics tool).

Performing exponential regression with these data points results in the model: $P(t) = 0.394 e^{0.126t}$ Where:

• $a \approx 0.394$,
• $b \approx 0.126$.

Thus, the exponential model is: $P(t) = 0.394 e^{0.126t}$

(b) Extrapolation to Estimate the Cost in 2008

To estimate the price of an apartment in 2008, we calculate $P(14)$, where $t = 14$ because 2008 is 14 years after 1994.

Using the exponential model $P(t) = 0.394 e^{0.126t}$, plug in $t = 14$:

$P(14) = 0.394 e^{0.126 \times 14}$

Let's compute this:

$P(14) = 0.394 \times e^{1.764}$ $P(14) \approx 0.394 \times 5.837$ $P(14) \approx 2.30 \, \text{million}$

Thus, the estimated price of a two-bedroom apartment in 2008 is approximately $2.3 million.

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Let me know if you would like more details or have any questions!

Here are 5 related questions for further exploration:

1. How would the model change if the price increase slowed after 2004?
2. What are the limitations of using an exponential model for real estate prices over long periods?
3. Can you apply a linear regression to this data, and how would the results compare?
4. How would the model behave if the growth rate $b$ was negative?
5. How sensitive is the exponential model to small changes in data points?

Tip: Exponential models are excellent for representing processes that grow at a constant percentage rate, like real estate prices or population growth!