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
y=44.65x+23,800; $172,600
y=44.65x+23,800;142,900
y=1488.33x+23,800;142,900
y=1488.33x23,800;172,600
The correct equation of the trend line that models the relationship between time and home prices is:
y = 1488.33x + 23,800
where y represents the median home price and x represents the year.
To predict the median home price in 2040, substitute x=2040 into the equation:
y = 1488.33(2040) + 23,800
y = 3,034,932 + 23,800
y = $3,058,732
Therefore, the predicted median home price in 2040 is $3,058,732.
To find the equation of the trend line that models the relationship between time and home prices, we can use the least squares method to find the line that best fits the data points.
First, let's assign the years (x-values) to the variable x and the prices (y-values) to the variable y:
x = [0, 10, 20, 30, 40, 50, 60]
y = [23800, 40900, 58000, 57400, 100700, 113100]
Next, we can calculate the slope and y-intercept of the trend line using the following formulas:
slope (m) = (n∑xy - ∑x∑y) / (n∑x^2 - (∑x)^2)
y-intercept (b) = (∑y - m∑x) / n
where n is the number of data points and ∑ denotes summation.
Calculating the necessary values:
n = 7
∑x = 0 + 10 + 20 + 30 + 40 + 50 + 60 = 210
∑y = 23800 + 40900 + 58000 + 57400 + 100700 + 113100 = 392400
∑xy = (0*23800) + (10*40900) + (20*58000) + (30*57400) + (40*100700) + (50*113100) = 65138000
∑x^2 = (0^2) + (10^2) + (20^2) + (30^2) + (40^2) + (50^2) + (60^2) = 15400
Now, let's substitute these values into the formulas:
m = (7*65138000 - 210*392400) / (7*15400 - 210^2)
b = (392400 - (44.65*210)) / 7
Simplifying these equations:
m = 1488.33
b ≈ 23800
Therefore, the equation of the trend line that models the relationship between time (x) and home prices (y) is:
y = 1488.33x + 23800
To predict the median home price in 2040 (which corresponds to x = 100), we can substitute x = 100 into the equation:
y = 1488.33(100) + 23800 = 148,833 + 23800 = $172,633
Therefore, the predicted median home price in 2040 is approximately $172,633.
To find the equation of a trend line that models the relationship between time (x) and home prices (y), we can use linear regression. Linear regression will help us find the best-fitting line that represents the data.
Step 1: Assign values to variables:
Let x represent the time (in years) and y represent the median home prices (in dollars).
Given data:
Year: 1940, 1950, 1960, 1970, 1980, 1990, 2000
Price: $23,800, $40,900, $58,000, $57,400, $100,700, $113,100
Let's assign the values to the variables:
x: 0, 10, 20, 30, 40, 50, 60
y: 23,800, 40,900, 58,000, 57,400, 100,700, 113,100
Step 2: Calculate the sums:
Next, calculate the sum of x, y, x^2, and xy.
Sum of x: Σx = 0 + 10 + 20 + 30 + 40 + 50 + 60 = 210
Sum of y: Σy = 23,800 + 40,900 + 58,000 + 57,400 + 100,700 + 113,100 = 393,900
Sum of x^2: Σx^2 = (0^2) + (10^2) + (20^2) + (30^2) + (40^2) + (50^2) + (60^2) = 14,700
Sum of xy: Σxy = (0 * 23,800) + (10 * 40,900) + (20 * 58,000) + (30 * 57,400) + (40 * 100,700) + (50 * 113,100) = 8,628,000
Step 3: Calculate the slope (m) of the trend line:
To find the slope (m), use the formula:
m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)
Where n is the number of data points, which in this case is 7.
m = (7 * 8,628,000 - (210 * 393,900)) / (7 * 14,700 - 210^2)
m = (60,396,000 - 82,719,000) / (102,900 - 44,100)
m = -22,323,000 / 58,800
m ≈ -379.73
Step 4: Calculate the y-intercept (b) of the trend line:
To find the y-intercept (b), use the formula:
b = (Σy - mΣx) / n
b = (393,900 - (-379.73 * 210)) / 7
b = 393,900 + 79,634.30 / 7
b ≈ 65,247.90 / 7
b ≈ 9,320.41
Step 5: Write the equation of the trend line:
Using the slope (m) and the y-intercept (b) we calculated, we can write the equation of the trend line in the form y = mx + b.
y = -379.73x + 9,320.41
Step 6: Predict the median home price in 2040:
To predict the median home price in 2040, substitute x = 100 into the equation.
y = -379.73(100) + 9,320.41
y ≈ -37,973 + 9,320.41
y ≈ -28,652.59
Based on the trend line equation, the predicted median home price in 2040 is approximately -$28,652.59. Note that this result is negative, which may not make logical sense. It could be due to extrapolating the data too far into the future, implying that other factors may influence the home prices beyond the available data range.