Answer:
a) Very strong positive correlation (r = 0.99).
b) y = 299.46 + 1.74x
c) $1,709 per month
Step-by-step explanation:
Linear regression is a statistical technique used to model the relationship between a dependent variable and an independent variable by fitting a linear equation to the observed data points. It aims to find the line of best fit that minimizes the overall distance between the observed data and the predicted values based on the linear equation.
Part (a)
We can use a statistical calculator to perform linear regression analysis on the given data.
The explanatory (independent) variable is the square footage (x-values).
The response (dependent) variable is monthly rent (y-values).
After entering the data into a statistical calculator we get:
- a = 299.464156...
- b = 1.73689309...
- r = 0.986575728...
The r-value, also known as the correlation coefficient, is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It ranges between -1 and 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.
In this case, an r-value of 0.99 suggests that there is a very strong positive linear relationship between between square footage and monthly rent. As the square footage increases, the monthly rent tends to increase as well.
This high correlation indicates that square footage is a strong predictor of monthly rent. However, it is important to note that the correlation coefficient alone does not imply causation or provide information about the underlying factors influencing the relationship.
[tex]\hrulefill[/tex]
Part (b)
To determine the regression equation, substitute the found values of a and b into the regression line formula, y = a + bx.
Round the values of a and b to the nearest hundredth:
Therefore, the regression equation is:
[tex]\boxed{y = 299.46 + 1.74x}[/tex]
where:
- x is the square footage.
- y is the monthly rent (in dollars).
[tex]\hrulefill[/tex]
Part (c)
To determine the monthly rent that we might expect to pay for an 810 square foot apartment, substitute x = 810 into the regression equation found in part (b), and solve for y:
[tex]\begin{aligned}y &= 299.46 + 1.74(810)\\&=299.46+1409.4\\&=1708.86\\&=1709\end{aligned}[/tex]
Therefore, we would expect to pay approximately $1,709 per month for an 810 square foot apartment.
