Using the data from the table, determine if there is a linear trend between the age of a house and its value and determine if there is an exact linear fit of the data. Describe the linear trend if there is one.
a. Negative linear trend, an exact linear fit.
b. Negative linear trend, not an exact linear fit.
c. Positive linear trend, an exact linear fit.
d. Positive linear trend, not an exact linear fit.
Answer (1)
Calculate the slopes between consecutive data points.
Observe that the slopes are not all equal, indicating that there is not an exact linear fit.
Note that the slopes are mostly positive, indicating a positive linear trend.
Conclude that there is a positive linear trend, but not an exact linear fit. d
Explanation
Understanding the Problem We are given a table of data showing the age of a house and its corresponding value. We need to determine if there is a linear trend between the age and value, and if the data has an exact linear fit.
Checking for Linear Trend To determine if there is a linear trend, we can calculate the slopes between consecutive data points. If the slopes are all equal, then there is an exact linear fit. If the slopes are generally increasing or decreasing, then there is a linear trend.
Calculating Slopes Let's calculate the slopes between consecutive data points:
Slope between (0, 140) and (3, 148): m 1 = 3 − 0 148 − 140 = 3 8 ≈ 2.67
Slope between (3, 148) and (6, 160): m 2 = 6 − 3 160 − 148 = 3 12 = 4
Slope between (6, 160) and (9, 162): m 3 = 9 − 6 162 − 160 = 3 2 ≈ 0.67
Slope between (9, 162) and (12, 185): m 4 = 12 − 9 185 − 162 = 3 23 ≈ 7.67
Slope between (12, 185) and (15, 178): m 5 = 15 − 12 178 − 185 = 3 − 7 ≈ − 2.33
Slope between (15, 178) and (18, 194): m 6 = 18 − 15 194 − 178 = 3 16 ≈ 5.33
Checking for Exact Linear Fit The slopes are approximately 2.67, 4, 0.67, 7.67, -2.33, and 5.33. Since the slopes are not all equal, there is not an exact linear fit.
Determining the Trend The slopes are mostly positive, but there is one negative slope. Overall, the values tend to increase as the age increases, so there is a positive linear trend.
Conclusion Therefore, the answer is positive linear trend, not an exact linear fit.
Examples
Understanding the relationship between a house's age and its value can be useful in real estate. For example, if you are looking to buy a house, you can use this information to estimate the value of a house based on its age. Similarly, if you are looking to sell a house, you can use this information to determine a fair selling price. This concept extends beyond real estate; it can be applied to assess the depreciation of cars, machinery, or any asset that loses or gains value over time. By analyzing trends, one can make informed decisions about investments, maintenance, and sales.
