When a relation appears to be linear in nature, a researcher will often wish to estimate this relationship between two variables with a straight line.

Regression Lines A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. A regression line is often used to predict values of Y for given values of X.

With correlation, an explanatory and response variable were not necessary. Usually in regression, there is an explanatory variable X and a response variable Y, but this is also unnecessary.

True Line The theory behind regression analysis relies on the assumption that there is a true line that describes the relationship between X and Y.

Not all points will lie exactly on the line due to the variation of variables. Instead, the line describes an equation for the average value of y for any given value of x, denoted as μy. The following is the equation for this theoretical line:

μy = β₀ + β₁x

β₀ is the true intercept and β₁ is the true slope.

Simple Linear Regression Model The following are three assumptions made when using the simple linear regression model:

1. For any given value of x, the response variable y follows a normal distribution. 2. Observed responses yᵢ are independent of each other. 3. The standard deviation σ of Y is the same for all values of X, and it is unknown.

From these assumptions, the simple linear regression equation is the following:

yᵢ = β₀ + β₁xᵢ + Ꮛᵢ

yᵢ is the response variable for the ith individual in the population.

β₀ and β₁ are parameters.

xᵢ is the value of the explanatory variable for the ith individual in the population.

Ꮛᵢ is a random error term, where Ꮛᵢ ~ N(0, σ).

Least Squares Regression Given a value of X, a researcher would like to predict the corresponding value of Y.

Unless the relationship is perfect, the value of y will never be exact.

A sample is used to estimate the equation of the true line.

ŷ = b₀ + bᵢx

This line is called the least squares regression line.

ŷ is the estimated value of μy for a given value of X.

The value of b₀ is called the intercept and estimates β₀. The value of b₁ is called the slope and estimates β₁.

A researcher would like to find the line that fits their data the best, and so they need to find the appropriate values of b₀ and b₁. There are infinitely many possible lines, and so, since they are using x to predict y, they would like a line that lies close to the points as possible in the vertical direction.

Therefore, the least squares regression line minimizes the sum of the squared deviations in the vertical direction.

Slope and Intercept The slope of the regression line b₁ is the predicted increase in y when x increases by one unit.

Another formula for b₁ that does not include correlation is the following:

The intercept of the regression line b₀ is the predicted value of y when x = 0.

Predicted Value of Y When the least squares regression line is determined, the predicted value of Y when X is equal to some value can be found using this equation.

Coefficient of Determination The coefficient of determination r² is the fraction of variation in Y that is accounted for by its regression on X.

If r = ±1, then r² = 1. Y can be predicted exactly for any value of X, since the regression on X accounts for all of the variation in Y.

If r = 0, then r² = 0. The regression on X says nothing about the value of y.

Other than these possible values, r² is in between 0 and 1.

Coefficient of Determination Interpretation The following is the general interpretation of the coefficient of determination:

"(100r²)% of the variation in (Y context) is accounted for by (X context)."

Consider the study of whether missing class affects a student's final grade. The following is the data of 14 STAT 1000 students with their total missed classes and final grades:

From this data, the least squares regression line is found to be ŷ = 82.84 – 2.24x and r² = 0.3716. a) Interpret this line. b) Interpret r². c) What would be the predicted final grade for a student who missed 7 classes?

a)

The slope b₁ = -2.24 indicates that, for each additional class missed, it can be predicted that a student's final grade will decrease by 2.24%.

The intercept b₀ = 82.84 indicates that, if a student does not miss any classes, the predicted final grade for that student would be 82.84%.

b)

37.16% of the variation in a student's final grade is accounted for by the number of classes they miss.

c)

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