Questions: Apartment Square Footage and Rent City Region County Square Footage, x Rent per Month, y Square Footage, x Rent per Month, y 500 650 1100 1875 588 1215 588 1075 1000 2000 1250 1775 688 1655 556 1050 825 1250 825 1300 460 1805 743 1475 1259 2700 660 1315 650 1200 975 1400 560 1250 1429 1900 1073 2350 800 1650 1452 3300 1906 4625 1305 3100 1077 1395

Solution Steps

To analyze the relationship between square footage and rent, we can use linear regression. This will help us understand how rent changes with square footage in both the city region and the county. We will use Python's libraries to perform linear regression and visualize the results.

We are given a table with two sets of data: one for a city region and another for a county. Each set contains square footage (\(x\)) and rent per month (\(y\)). We need to analyze this data, but the specific question or task is not provided. For the sake of this exercise, let's assume we need to find the linear relationship between square footage and rent for the city region.

Let's list the data points for the city region:

• (500, 650)
• (588, 1215)
• (1000, 2000)
• (688, 1655)
• (825, 1250)
• (460, 1805)
• (1259, 2700)
• (650, 1200)
• (560, 1250)
• (1073, 2350)
• (1452, 3300)
• (1305, 3100)

To find the linear relationship, we will use the formula for the line of best fit (linear regression), which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

The formula for the slope \(m\) is:

\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]

Where:

• \(n\) is the number of data points.
• \(\sum xy\) is the sum of the product of each \(x\) and \(y\).
• \(\sum x\) is the sum of all \(x\) values.
• \(\sum y\) is the sum of all \(y\) values.
• \(\sum x^2\) is the sum of each \(x\) squared.

The formula for the y-intercept \(b\) is:

\[ b = \frac{\sum y - m(\sum x)}{n} \]

Let's calculate the necessary sums for the city region data:

• \(\sum x = 500 + 588 + 1000 + 688 + 825 + 460 + 1259 + 650 + 560 + 1073 + 1452 + 1305 = 11360\)
• \(\sum y = 650 + 1215 + 2000 + 1655 + 1250 + 1805 + 2700 + 1200 + 1250 + 2350 + 3300 + 3100 = 23275\)
• \(\sum xy = (500 \times 650) + (588 \times 1215) + (1000 \times 2000) + (688 \times 1655) + (825 \times 1250) + (460 \times 1805) + (1259 \times 2700) + (650 \times 1200) + (560 \times 1250) + (1073 \times 2350) + (1452 \times 3300) + (1305 \times 3100) = 26375550\)
• \(\sum x^2 = 500^2 + 588^2 + 1000^2 + 688^2 + 825^2 + 460^2 + 1259^2 + 650^2 + 560^2 + 1073^2 + 1452^2 + 1305^2 = 13009154\)

Now, substitute these sums into the formulas for \(m\) and \(b\):

\[ m = \frac{12(26375550) - (11360)(23275)}{12(13009154) - (11360)^2} \]

\[ b = \frac{23275 - m(11360)}{12} \]

\[ m = \frac{316506600 - 264740000}{156109848 - 129049600} = \frac{51866600}{27060248} \approx 1.916 \]

\[ b = \frac{23275 - 1.916 \times 11360}{12} = \frac{23275 - 21761.76}{12} \approx 125.6867 \]

Final Answer