Question
Time (year) Value of House
1 157,200
2 161,000
3 173,000
4 178,000
5 190,000
Heather and Joel bought a house for $157,200 and know that the house appreciates every year. They keep track of their house value for 5 years and want to find a regression equation to predict how much it will be worth in 8 years. Which equation is the BEST model for the data?
Responses
A y = -1170
+ 17500x + 125000y = -1170 x 2 + 17500x + 125000
B y = 150000
y = 150000 ( 1.004 ) 12 t
C y = 150000
y = 150000 ( 1.006 ) 12 t
D y = -1166.67
+ 17833.33x + 130000
Answers
GPT-4o mini
To determine the best regression equation for modeling the house value data over the given 5-year period, we need to evaluate the provided options based on their structure and the type of data we have.
The data indicates the yearly increase in the value of the house, which suggests that the value is likely increasing, potentially at a constant rate. Looking at the growth, we can determine if it is linear or exponential.
Let's briefly analyze the given options:
**A**: \( y = -1170 + 17500x + 125000 \)
This represents a polynomial model that is less likely to fit the data well since it does not align with the observed growth pattern.
**B**: \( y = 150000 (1.004)^{12t} \)
This suggests exponential growth, but the base \(1.004\) implies a very slow growth rate. It doesn’t fit the data.
**C**: \( y = 150000 (1.006)^{12t} \)
This too suggests exponential growth, and \(1.006\) indicates a moderately slow growth rate as well. Still, it does not seem to fit our 5-year observed data closely.
**D**: \( y = -1166.67 + 17833.33x + 130000 \)
This again suggests a polynomial form, which does not seem appropriate for the manageable increase in house value.
Since we need a prediction model for the data, if we examined the given data points' values for any trends, we see that the value of houses increases each year.
Calculating the average increase over the 5 years seems the right path to take. The average increase from year to year can be computed as follows:
- Year 2 - Year 1: \( 161,000 - 157,200 = 1,800 \)
- Year 3 - Year 2: \( 173,000 - 161,000 = 12,000 \)
- Year 4 - Year 3: \( 178,000 - 173,000 = 5,000 \)
- Year 5 - Year 4: \( 190,000 - 178,000 = 12,000 \)
Thus, a rough estimate would be that the house appreciates consistently by an average amount just over 5,000 per year. In 8 years, we would look to calculate from the 5th year’s value.
None of the options match a simple linear model directly from the calculated appreciation, however, among the candidates, B and C are the formulas that suggest a trend onwards, while A and D add more complexity.
Without exact regression calculations from the data, the safest approximate would lean toward a model that retains growth through linearity or common exponentials—likely leading to an exponential response as seen in B or C.
Hence, the most suitable equation among the choices that might model the appreciation of house value better than polynomials would most likely be:
**C**: \( y = 150000 (1.006)^{12t} \)
However, please verify whether the appreciation needs time input values distinctly or if calculated yearly too before final observance.
The data indicates the yearly increase in the value of the house, which suggests that the value is likely increasing, potentially at a constant rate. Looking at the growth, we can determine if it is linear or exponential.
Let's briefly analyze the given options:
**A**: \( y = -1170 + 17500x + 125000 \)
This represents a polynomial model that is less likely to fit the data well since it does not align with the observed growth pattern.
**B**: \( y = 150000 (1.004)^{12t} \)
This suggests exponential growth, but the base \(1.004\) implies a very slow growth rate. It doesn’t fit the data.
**C**: \( y = 150000 (1.006)^{12t} \)
This too suggests exponential growth, and \(1.006\) indicates a moderately slow growth rate as well. Still, it does not seem to fit our 5-year observed data closely.
**D**: \( y = -1166.67 + 17833.33x + 130000 \)
This again suggests a polynomial form, which does not seem appropriate for the manageable increase in house value.
Since we need a prediction model for the data, if we examined the given data points' values for any trends, we see that the value of houses increases each year.
Calculating the average increase over the 5 years seems the right path to take. The average increase from year to year can be computed as follows:
- Year 2 - Year 1: \( 161,000 - 157,200 = 1,800 \)
- Year 3 - Year 2: \( 173,000 - 161,000 = 12,000 \)
- Year 4 - Year 3: \( 178,000 - 173,000 = 5,000 \)
- Year 5 - Year 4: \( 190,000 - 178,000 = 12,000 \)
Thus, a rough estimate would be that the house appreciates consistently by an average amount just over 5,000 per year. In 8 years, we would look to calculate from the 5th year’s value.
None of the options match a simple linear model directly from the calculated appreciation, however, among the candidates, B and C are the formulas that suggest a trend onwards, while A and D add more complexity.
Without exact regression calculations from the data, the safest approximate would lean toward a model that retains growth through linearity or common exponentials—likely leading to an exponential response as seen in B or C.
Hence, the most suitable equation among the choices that might model the appreciation of house value better than polynomials would most likely be:
**C**: \( y = 150000 (1.006)^{12t} \)
However, please verify whether the appreciation needs time input values distinctly or if calculated yearly too before final observance.