Suppose that a simple linear regression model is appropriate...

Suppose that a simple linear regression model is appropriate for describing the relationship between $y=$ house price (in dollars) and $x=$ house size (in square feet) for houses in a large city. The population regression line is $y=23,000+47 x$ and $\sigma_{e}=5000$ a. What is the average change in price associated with one extra square foot of space? With an additional 100 sq. $f t$. of space? b. Approximately what proportion of 1800 sq. ft. homes would be priced over $\$ 110,000 ?$ Under $\$ 100,000 ?$

Suppose that a simple linear regression model is appropriate for describing the relationship between $y=$ house price (in dollars) and $x=$ house size (in square feet) for houses in a large city. The population regression line is $y=23,000+47 x$ and $\sigma_{e}=5000$
a. What is the average change in price associated with one extra square foot of space? With an additional 100 sq. $f t$. of space?
b. Approximately what proportion of 1800 sq. ft. homes would be priced over $\$ 110,000 ?$ Under $\$ 100,000 ?$

Key Concepts

Simple Linear Regression

Simple linear regression is a statistical method used to model the relationship between a dependent variable and one independent variable with a linear equation. This method estimates an equation that best fits the data by minimizing the sum of squared differences between the observed and predicted values.

Interpretation of the Slope

The slope in a simple linear regression indicates the average change in the dependent variable for a one-unit increase in the independent variable. It is a measure of the strength and direction of the association between the two variables, and it allows for easy interpretation of how changes in the independent variable are expected to affect the dependent variable.

Prediction Using the Regression Model

Once the regression line is established, predictions about the dependent variable can be made for specific values of the independent variable. This involves plugging a given value into the regression equation to estimate the corresponding value of the dependent variable.

Error Term and Its Standard Deviation

The error term in a regression model represents the difference between the actual observed values and those predicted by the model. Its standard deviation, often symbolized as sigma, quantifies the typical size of these errors and reflects the degree of variability in the observations that is not explained by the regression model.

Normal Distribution in Regression Inference

When the error terms in a regression model are assumed to be normally distributed, this property can be used to make probabilistic statements about the dependent variable. Standardizing the differences between observed and predicted values allows for the calculation of z-scores, which in turn enable the determination of probabilities associated with various outcomes.

Transcript

00:02 All right, so we're looking at a regression analysis relating between market value and dollars to the size and square foot footage of homes in tennessee.

00:15 Here's the equation.

00:16 The value is negative 37 ,136 plus 65 times the size and square footage.

00:29 And we have these, this, this, this anova here, as well as some t.

00:39 Values some other information here and so how many homes in the sample well there were 35 because there are 34 degrees of freedom total so degrees of freedom plus one is the sample size standard error of the estimate is given it's right here this value so it's the square root of the sum of the squares the sum of the squares of the error so divided by n minus two equals the sum of squares of the error.

01:15 This value divided by 33.

01:19 That's the standard error of the estimate.

01:24 And we're getting some large sum of squares, so that's a pretty reasonable value to get.

01:28 5 ,456, round it, 54, 57.

01:34 The coefficient of determination.

01:38 A few ways we can get it here.

01:40 The sum of squares of the residual with our regression.

01:45 Over the total, or one minus the sum, squares of the error, divided by the total.

01:51 Either way, it doesn't matter.

01:52 We'll get the same thing.

01:54 But i'll do the first one, the regression, sum of the squares of the regression, divided by the total.

02:02 Coefficient determination, 0 .93.