Year	1	3	6	7	10
Population	12,811	13,068	13,421	13,516	13,799
 
Determine a model for the given data by using quadratic regression.
What will be the population in year 14?

To determine a model using quadratic regression and predict the population for year 14, we follow these steps:

### Step 1: Define the quadratic model
The general form of a quadratic model is:

\[
P(t) = at^2 + bt + c
\]

where:
- \( t \) represents the year,
- \( P(t) \) represents the population at year \( t \),
- \( a \), \( b \), and \( c \) are constants to be determined from the data.

### Step 2: Translate the years
To simplify calculations, let's translate the years so that the first year corresponds to \( t = 0 \). Subtract 1 from all years in the data.

- Year 1 → \( t = 0 \)
- Year 3 → \( t = 2 \)
- Year 6 → \( t = 5 \)
- Year 7 → \( t = 6 \)
- Year 10 → \( t = 9 \)

The data now becomes:
- \( (0, 12811) \)
- \( (2, 13068) \)
- \( (5, 13421) \)
- \( (6, 13516) \)
- \( (9, 13799) \)

### Step 3: Perform quadratic regression
We now apply quadratic regression to fit a model of the form \( P(t) = at^2 + bt + c \). I'll compute this regression to find the values of \( a \), \( b \), and \( c \).

Let me calculate this for you.The quadratic regression model for the population data is:

\[
P(t) = -2.83t^2 + 135.27t + 12810.49
\]

where \( t \) represents the years starting from year 1 (i.e., \( t = 0 \) corresponds to year 1, \( t = 2 \) corresponds to year 3, and so on).

Using this model, the predicted population for year 14 (which corresponds to \( t = 13 \)) is approximately:

\[
P(14) \approx 14,090.53
\]

### Would you like more details on the regression process or have any other questions?

Here are 5 related questions to expand on this:
1. What does each coefficient in the quadratic equation represent?
2. How would the population change if we used linear regression instead of quadratic?
3. Can we use this model to predict populations beyond 20 years?
4. How does quadratic regression differ from exponential regression for population growth?
5. What are potential sources of error in using this model for long-term predictions?

**Tip:** When using regression models for predictions far outside the data range, be cautious, as models may not capture long-term trends accurately.

Year 1 3 6 7 10
Population 12,811 13,068 13,421 13,516 13,799

Determine a model for the given data by using quadratic regression. What will be the population in year 14?

Learn how to determine a quadratic regression model for population data. The given data includes populations for years 1, 3, 6, 7, and 10. Using the regression model, predict the population for year 14. Suitable for Grades 11-12 and covers key concepts like polynomial fitting and predictive modeling.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation