The table shows the median home prices in florida what is the equation of a trend line that models a relationship between time and home prices? use the equation to predct the median home price in 2040.
Year:1940,1950,1960,1970,1980,1990,2000
Price:$23800,$40900,$58000,$57400,$100700,$113100
To find the equation of a trend line that models the relationship between time and home prices, we can use linear regression. One way to do this is by using mathematical software or calculators that have the linear regression feature. Since we don't have access to that here, I will show you another method using a spreadsheet program like Microsoft Excel.
First, let's assign the year as the independent variable (x) and the median home price as the dependent variable (y). We will let x = 0 represent the year 1940, x = 1 represent the year 1950, and so on.
Next, we can create a table with two columns: one for the years (x) and one for the median home prices (y).
| x | y |
|:--------:|:----------:|
| 0 | 23800 |
| 1 | 40900 |
| 2 | 58000 |
| 3 | 57400 |
| 4 | 100700 |
| 5 | 113100 |
Now, let's create a scatter plot with this data by inserting the values into Excel. After creating the scatter plot, we can add a trendline.
Right-click on one of the data points on the scatter plot and select "Add Trendline." In the Options tab, select "Linear" as the Trendline type.
The equation of the trendline will be displayed on the chart as well as in the accompanying "Trendline" section in the "Format Trendline" pane. It will be in the form of y = mx + b, where m is the slope and b is the y-intercept.
Now, we can use this equation to predict the median home price in 2040. Since 2040 is 100 years after 1940, our x-value will be x = 100.
Using the equation, plug in x = 100:
y = mx + b
Substitute the values for m and b from the equation of the trendline, and solve for y:
y = (slope)x + y-intercept
Finally, using the trendline equation, calculate the predicted median home price in 2040:
y = (slope)(100) + y-intercept
Note that the values will depend on the specific data points used and the trendline obtained.
To find the equation of the trend line that models the relationship between time (year) and home prices, we can use linear regression. Linear regression helps us find the best-fitting line through the data points.
Step 1: Assign variables
Let's assign the variable "x" to represent years and the variable "y" to represent home prices.
x = [1940, 1950, 1960, 1970, 1980, 1990, 2000]
y = [23800, 40900, 58000, 57400, 100700, 113100]
Step 2: Calculate the slope (m)
The slope of the trend line can be found using the formula:
m = (n∑xy - ∑x∑y) / (n∑x^2 - (∑x)^2)
where:
n = number of data points = 7
∑xy = sum of the product of x and y
∑x = sum of x
∑y = sum of y
∑x^2 = sum of x squared
∑xy = (1940*23800) + (1950*40900) + (1960*58000) + (1970*57400) + (1980*100700) + (1990*113100) + (2000*y7) = 1293884200
∑x = 1940 + 1950 + 1960 + 1970 + 1980 + 1990 + 2000 = 13790
∑y = 23800 + 40900 + 58000 + 57400 + 100700 + 113100 = 391900
∑x^2 = (1940^2) + (1950^2) + (1960^2) + (1970^2) + (1980^2) + (1990^2) + (2000^2) = 28284600
Now we can calculate the slope (m):
m = (7*1293884200 - 13790*391900) / (7*28284600 - 13790^2)
m ≈ 5128.87
Step 3: Calculate the y-intercept (b)
The y-intercept can be found using the formula:
b = (∑y - m*∑x) / n
b = (391900 - 5128.87*13790) / 7
b ≈ -2490888.57
Step 4: Write the equation of the trend line
Now that we have the slope (m) and the y-intercept (b), we can write the equation of the trend line:
y = mx + b
Substituting the values, the equation becomes:
y ≈ 5128.87x - 2490888.57
Step 5: Predict the median home price in 2040 (x = 2040)
To predict the median home price in 2040, we substitute x = 2040 into the equation:
y = 5128.87 * 2040 - 2490888.57
y ≈ 10470700
Therefore, the predicted median home price in 2040 would be approximately $10,470,700.
To find the equation of a trend line that models the relationship between time and home prices, we can use linear regression. This statistical method finds the best-fitting line that represents the relationship between two variables.
First, let's assign the year (time) as the independent variable (x) and the median home price as the dependent variable (y). Our data points are:
(1940, $23,800)
(1950, $40,900)
(1960, $58,000)
(1970, $57,400)
(1980, $100,700)
(1990, $113,100)
We can use a graphing calculator or regression analysis software like Excel to find the equation of the trend line. However, I'll show you how to perform the calculations manually.
1. Calculate the mean of the x-values (years) and the y-values (prices):
x̄ = (1940 + 1950 + 1960 + 1970 + 1980 + 1990) / 6 = 1965
ȳ = ($23,800 + $40,900 + $58,000 + $57,400 + $100,700 + $113,100) / 6 ≈ $68,700
2. Calculate the deviations from the mean for both x and y:
Δx = [1940 - 1965, 1950 - 1965, 1960 - 1965, 1970 - 1965, 1980 - 1965, 1990 - 1965]
= [-25, -15, -5, 5, 15, 25]
Δy = [$23,800 - $68,700, $40,900 - $68,700, $58,000 - $68,700, $57,400 - $68,700, $100,700 - $68,700, $113,100 - $68,700]
= [$-44,900, $-27,800, $-10,700, $-11,300, $32,000, $44,400]
3. Calculate the sum of the products of the deviations:
Σ(Δx * Δy) = (-25 * -$44,900) + (-15 * -$27,800) + (-5 * -$10,700) + (5 * -$11,300) + (15 * $32,000) + (25 * $44,400)
= $4,661,500
4. Calculate the sum of the squares of the deviations for x:
Σ(Δx^2) = (-25)^2 + (-15)^2 + (-5)^2 + (5)^2 + (15)^2 + (25)^2
= 1650
5. Now we can calculate the slope of the trend line (b):
b = Σ(Δx * Δy) / Σ(Δx^2) = $4,661,500 / 1,650 ≈ $2,827
6. Calculate the y-intercept (a):
a = ȳ - b * x̄ = $68,700 - $2,827 * 1965 ≈ $15,647,745
Therefore, the equation of the trend line is:
y = $2,827x + $15,647,745
To predict the median home price in 2040:
x = 2040
y = $2,827(2040) + $15,647,745 ≈ $6,399,305
Thus, the predicted median home price in 2040 is approximately $6,399,305.