Topics
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Best fit lines can also be drawn approximately by eye. We start by finding the average ​x​ and ​y, giving the point ​(xˉ,yˉ​). We then take a ruler and place it on this point, and adjust the slope until we find a reasonable best fit line.
Linear regression is a statistical method used to model the relationship between two variables when data is given as pairs of points ​(x,y). We fit a straight line (called the regression line) that minimizes the average vertical distance from the points:
The general equation of the regression line is:
where ​a​ is the slope and ​b​ is the ​y​-intercept.
The values of ​a​ and ​b​ can be found using a calculator:
Use Stat>Edit to fill in ​x​- and ​y​-values into ​L1​​ and ​L2​.
Then, press Stat, right arrow to the CALC menu, and select 4:LinReg(ax+b).
Pearson's product-moment correlation coefficient, denoted by ​r, measures the strength and direction of a linear relationship between two numerical variables ​x​ and ​y. Its value always lies between ​−1​ and ​+1:
​r=+1: perfect positive linear relationship
​r=−1: perfect negative linear relationship
​r=0: no linear relationship
A positive value means ​y​ generally increases as ​x​ increases; a negative value means ​y​ generally decreases as ​x​ increases. The closer ​r​ is to ​±1, the stronger the linear relationship.
If you clickmode, scroll to STAT DIAGNOSTICS , hover over ON, and click ENTER, then any time you perform a linear regression, the calculator will provide Pearson's coefficient in addition to the regression line.
Once we have a regression line ​y=ax+b, we can use it to predict ​y​ by plugging in a value of ​x.
When using a regression line to predict ​y​ from ​x, we need to be aware of the danger of extrapolation. This occurs when we try to predict ​y​ for a value of ​x​ far outside the range of ​x​ values in our data. For such an ​x, we cannot trust that the relationship is the same.
While it is possible to use a regression line ​y=ax+b​ to predict ​x​ with
this is not a reliable process. The best fit line is determined to minimize the difference between the real ​y’s​ and the predicted ​y’s,​so the difference between real and predicted values for ​x​ may be much larger.
In the same way that we can plot a straight line minimizing the vertical distances from points ​(x,y), we can plot a straight line minimizing the horizontal distances. This is called an ​x​ on ​y​ regression line. We calculate an ​x​ on ​y​ regression line by switching our ​x​ and ​y​ lists while using LinReg(ax+b).
With this line we can make reliable predictions of ​x​ given ​y, so long as we are not extrpolating.
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Best fit lines can also be drawn approximately by eye. We start by finding the average ​x​ and ​y, giving the point ​(xˉ,yˉ​). We then take a ruler and place it on this point, and adjust the slope until we find a reasonable best fit line.
Linear regression is a statistical method used to model the relationship between two variables when data is given as pairs of points ​(x,y). We fit a straight line (called the regression line) that minimizes the average vertical distance from the points:
The general equation of the regression line is:
where ​a​ is the slope and ​b​ is the ​y​-intercept.
The values of ​a​ and ​b​ can be found using a calculator:
Use Stat>Edit to fill in ​x​- and ​y​-values into ​L1​​ and ​L2​.
Then, press Stat, right arrow to the CALC menu, and select 4:LinReg(ax+b).
Pearson's product-moment correlation coefficient, denoted by ​r, measures the strength and direction of a linear relationship between two numerical variables ​x​ and ​y. Its value always lies between ​−1​ and ​+1:
​r=+1: perfect positive linear relationship
​r=−1: perfect negative linear relationship
​r=0: no linear relationship
A positive value means ​y​ generally increases as ​x​ increases; a negative value means ​y​ generally decreases as ​x​ increases. The closer ​r​ is to ​±1, the stronger the linear relationship.
If you clickmode, scroll to STAT DIAGNOSTICS , hover over ON, and click ENTER, then any time you perform a linear regression, the calculator will provide Pearson's coefficient in addition to the regression line.
Once we have a regression line ​y=ax+b, we can use it to predict ​y​ by plugging in a value of ​x.
When using a regression line to predict ​y​ from ​x, we need to be aware of the danger of extrapolation. This occurs when we try to predict ​y​ for a value of ​x​ far outside the range of ​x​ values in our data. For such an ​x, we cannot trust that the relationship is the same.
While it is possible to use a regression line ​y=ax+b​ to predict ​x​ with
this is not a reliable process. The best fit line is determined to minimize the difference between the real ​y’s​ and the predicted ​y’s,​so the difference between real and predicted values for ​x​ may be much larger.
In the same way that we can plot a straight line minimizing the vertical distances from points ​(x,y), we can plot a straight line minimizing the horizontal distances. This is called an ​x​ on ​y​ regression line. We calculate an ​x​ on ​y​ regression line by switching our ​x​ and ​y​ lists while using LinReg(ax+b).
With this line we can make reliable predictions of ​x​ given ​y, so long as we are not extrpolating.