8. Two Variable Data Analysis

Lesson

Once we have a least squares line, we're ready to start making predictions. Since our line is the best possible fit for the data we have, we can use it as a model to predict the likely value for the dependent variable based on a value for the independent variable or vice versa.

For example:

- Given a value for the independent variable, $x$
`x`, we can substitute it into the equation to find a value for the dependent variable, $y$`y`. - Likewise, given a value for the dependent variable, $y$
`y`, we can substitute it into the equation to find a value for the independent variable $x$`x`.

If the value used is within the range of data, this type of prediction is called interpolation. However, if the value used lies outside the range of data, then this is called extrapolation. The further outside the range of data your chosen value is, the less reliable the prediction.

This next video will demonstrate how we can make a prediction once we've input our data and calculated the least squares regression line.

The slope of a least squares line, tells us the average rate of change of one variable with respect to another variable. We usually say that the dependent variable, increases/decreases for each unit of the independent variable.

The $y$`y`-intercept tells us what the dependent variable is predicted to be when the independent variable is $0$0.

A least squares regression line is given by $y=3.59x+6.72$`y`=3.59`x`+6.72.

State the slope of the line.

Which of the following is true?

The slope of the line indicates that the bivariate data set has a positive correlation.

AThe slope of the line indicates that the bivariate data set has a negative correlation.

BThe slope of the line indicates that the bivariate data set has a positive correlation.

AThe slope of the line indicates that the bivariate data set has a negative correlation.

BWhich of the following is true?

If $x$

`x`increases by $1$1 unit, then $y$`y`increases by $3.59$3.59 units.AIf $x$

`x`increases by $1$1 unit, then $y$`y`decreases by $3.59$3.59 units.BIf $x$

`x`increases by $1$1 unit, then $y$`y`decreases by $6.72$6.72 units.CIf $x$

`x`increases by $1$1 unit, then $y$`y`increases by $6.72$6.72 units.DIf $x$

`x`increases by $1$1 unit, then $y$`y`increases by $3.59$3.59 units.AIf $x$

`x`increases by $1$1 unit, then $y$`y`decreases by $3.59$3.59 units.BIf $x$

`x`increases by $1$1 unit, then $y$`y`decreases by $6.72$6.72 units.CIf $x$

`x`increases by $1$1 unit, then $y$`y`increases by $6.72$6.72 units.DState the value of the $y$

`y`-intercept.

Create a scatter plot to represent the relationship between two variables, determine the correlation between these variables by testing different regression models using technology, and use a model to make predictions when appropriate.

Describe the value of mathematical modelling and how it is used in real life to inform decisions.