Question

the table shows the relationship between square footage and rent prices for different locations.
what does the slope say about the data? use the linear regression calculator to find the slope.
the slope indicates that rent doubles as square footage increases, reflecting a high premium on larger properties.
the slope indicates no clear trend between square footage and rent, suggesting that factors other than size may influence rent costs.
the slope indicates that as the square footage increases, the rent increases proportionally, demonstrating a direct correlation between size and cost.

Explanation:

Step1: Calculate the slope

The formula for the slope \(m\) of a line in linear regression (using two - point formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\)). Let's take two points \((x_1,y_1)=(600,850)\) and \((x_2,y_2)=(800,1300)\).

$$m=\frac{1300 - 850}{800 - 600}=\frac{450}{200}=2.25$$

Let's check with another pair of points \((x_1,y_1)=(800,1300)\) and \((x_2,y_2)=(1000,1600)\)

$$m=\frac{1600 - 1300}{1000 - 800}=\frac{300}{200}=1.5$$

Using the linear regression formula (assuming we use a calculator for all three points \((x_1,y_1)=(600,850)\), \((x_2,y_2)=(800,1300)\), \((x_3,y_3)=(1000,1600)\)):
The general formula for the slope of the least - squares regression line is \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\)
Here \(n = 3\), \(\sum_{i=1}^{3}x_i=600 + 800+1000=2400\), \(\sum_{i = 1}^{3}y_i=850 + 1300+1600=3750\), \(\sum_{i=1}^{3}x_iy_i=(600\times850)+(800\times1300)+(1000\times1600)=510000+1040000 + 1600000=3150000\), \(\sum_{i = 1}^{3}x_i^{2}=600^{2}+800^{2}+1000^{2}=360000+640000+1000000 = 2000000\)

$$m=\frac{3\times3150000-2400\times3750}{3\times2000000-(2400)^{2}}=\frac{9450000 - 9000000}{6000000 - 5760000}=\frac{450000}{240000}=1.875$$

Step2: Interpret the slope

The slope \(m = 1.875\) means that for every additional square foot of space, the rent increases by approximately \(\$1.88\).

Answer:

The slope indicates that as the square footage increases, the rent increases proportionally, demonstrating a direct correlation between size and cost.