Given the following data for four houses sold in comparable neighborhoods and their corresponding number of square feet, draw a scatter plot and find a linear regression equation representing a relationship between the number of square feet and the selling price of the house.
Square Feet 1527 2549 3002 3018
Selling Price $92,903 $212,000 $251,000 $268,000
Here is the scatter plot representing the relationship between the number of square feet and the selling price of the house:
```
Square Feet | | |
Selling Price |-|----|-------
0 1000 2000 3000
```
To find the linear regression equation representing the relationship between the number of square feet and the selling price of the house, we can use statistical software or a calculator that has a regression function. This will provide us with the equation in the form of:
Selling Price = Slope * Square Feet + Intercept
The regression equation for this data is as follows:
Selling Price = 60.79 * Square Feet + 5,899.25
are you sure
Apologies for the previous incorrect response. I made an error in calculating the regression equation. Let me recalculate it and provide you with the correct answer.
First, let's organize the data:
Square Feet: 1527, 2549, 3002, 3018
Selling Price: $92,903, $212,000, $251,000, $268,000
Now, let's plot the scatter plot:
```
Selling Price (in thousands)
|
| .
| .
| .
| .
|______________________
0 1000 2000 3000 Square Feet
```
To find the linear regression equation, we need to calculate the slope (m) and the y-intercept (b). We can do this using the equations:
m = (n∑(XY) - (∑X)(∑Y)) / (n∑(X^2) - (∑X)^2)
b = (∑Y - m(∑X)) / n
Where n is the number of data points, ∑X is the sum of the X values, ∑Y is the sum of the Y values, ∑(XY) is the sum of the products of X and Y, and ∑(X^2) is the sum of the squared X values.
Calculating the values:
n = 4
∑X = 1527 + 2549 + 3002 + 3018 = 10096
∑Y = 92903 + 212000 + 251000 + 268000 = 824903
∑(XY) = (1527 * 92903) + (2549 * 212000) + (3002 * 251000) + (3018 * 268000) = 1997870310
∑(X^2) = (1527^2) + (2549^2) + (3002^2) + (3018^2) = 22813968
Now we can calculate the slope (m):
m = (4 * 1997870310 - (10096 * 824903)) / (4 * 22813968 - (10096^2))
m = (7991481240 - 8264661088) / (91255872 - 101932416)
m = -273179848 / -106767544
m ≈ 2.56
And finally, the y-intercept (b):
b = (824903 - (2.56 * 10096)) / 4
b = (824903 - 25871.36) / 4
b ≈ 199758.16
Therefore, the linear regression equation representing the relationship between the number of square feet and the selling price of the house is:
Selling Price = 2.56 * Square Feet + 199758.16
In the United States, the revenue (money taken from sales) at a "full service" restaurant is increasing at a faster rate than the revenue at a "fast food" restaurant. The data below represent the annual revenue in billions of dollars for each type of restaurant. Use the data to answer the three questions below. Let x = 0 represent the year 1990.
Apologies, but it seems that the data you mentioned is missing. Could you please provide the annual revenue values for each type of restaurant?
To draw a scatter plot and find a linear regression equation, follow these steps:
1. Start by organizing the given data into two columns: one for the number of square feet and the other for the selling price.
Number of Square Feet: 1527, 2549, 3002, 3018
Selling Price: $92,903, $212,000, $251,000, $268,000
2. Plot each data point on a scatter plot with the number of square feet on the x-axis and the selling price on the y-axis.
Here is an example visual representation of the scatter plot:
Selling Price
|
$300,000 -
|
$250,000 -
|
$200,000 -
|
$150,000 -
|________________________
1000 2000 3000 4000 Number of Square Feet
3. Once the scatter plot is completed, draw the best-fit line that represents the linear regression equation.
The best-fit line should go through the scatter points in such a way that it represents the general trend or pattern of the data points. This line represents the equation that best fits the given data.
4. To find the equation of the best-fit line (linear regression equation), you can use statistical software, spreadsheet software, or manually calculate it using the least squares method.
Using statistical or spreadsheet software:
- Input the data points into the software.
- Perform a linear regression analysis.
- The output will provide you with the equation of the best-fit line, which will include the y-intercept (b₀) and the slope (b₁).
Manually calculating the linear regression equation using the least squares method:
- Find the means of both the x-values and the y-values (mean of the number of square feet and mean of the selling price).
- Calculate the deviations from the means for both x and y.
- Calculate the sum of the products of the deviations.
- Calculate the sum of the squared deviations for x.
- Calculate the slope (b₁) by dividing the sum of the products of deviations by the sum of squared deviations for x.
- Calculate the y-intercept (b₀) by substituting the mean values and the slope into the equation: b₀ = mean(y) - b₁ * mean(x).
Once you have the linear regression equation (y = b₀ + b₁x), you can use it to predict the selling price (y) based on a given number of square feet (x).