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AP Statistics : How to find the least-squares regression line

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Example Question #1 : How To Find The Least Squares Regression Line

In a regression analysis, the y-variable should be the ___________ variable, and the x-variable should be the ___________ variable.

Possible Answers:

Qualified, Unqualified

First, Second

Greater, Lesser

Dependent, Independent

Independent, Dependent

Correct answer:

Dependent, Independent

Explanation:

Regression tests seek to determine one variable's ability to predict another variable. In this analysis, one variable is dependent (the one predicted), and the other is independent (the variable that predicts). Therefore, the dependent variable is the y-variable and the independent variable is the x-variable.

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Example Question #1 : How To Find The Least Squares Regression Line

If a data set has a perfect negative linear correlation, has a slope of $\displaystyle -7$ and an explanatory variable standard deviation of $\displaystyle 2$, what is the standard deviation of the response variable?

Possible Answers:

$\displaystyle -14$

$\displaystyle 0$

$\displaystyle 2$

$\displaystyle 14$

$\displaystyle 7$

Correct answer:

$\displaystyle 14$

Explanation:

The key here is to utilize

$b_{1} = r\frac{s_{y}}{s_{x}}$ $\displaystyle b_{1} = r\frac{s_{y}}{s_{x}}$.

"Perfect negative linear correlation" means r = -1 $\displaystyle r = -1$, while the rest of the problem indicates $b_{1} = -7$ $\displaystyle b_{1} = -7$ and $s_{x} = 2$ $\displaystyle s_{x} = 2$. This enables us to solve for $s_{y}$ $\displaystyle s_{y}$.

$b_{1} = r\frac{s_{y}}{s_{x}}$ $\displaystyle b_{1} = r\frac{s_{y}}{s_{x}}$
$\frac{b_{1}}{r} = \frac{s_{y}}{s_{x}}$ $\displaystyle \frac{b_{1}}{r} = \frac{s_{y}}{s_{x}}$
$s_{y} = \frac{b_{1}}{r}{s_{x}} = \frac{-7}{-1}2 = 7\cdot 2 = 14$ $\displaystyle s_{y} = \frac{b_{1}}{r}{s_{x}} = \frac{-7}{-1}2 = 7\cdot 2 = 14$

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Example Question #4 : Bivariate Data

A least-squares regression line has equation $\displaystyle y = 0.64x + 8$ and a correlation of r = 0.8 $\displaystyle r = 0.8$. It is also known that s_x = 5 $\displaystyle s_x = 5$. What is s_y? $\displaystyle s_y?$

Possible Answers:

$\displaystyle 14$

$\displaystyle 15$

$\displaystyle 16$

$\displaystyle 4$

$\displaystyle 5$

Correct answer:

$\displaystyle 4$

Explanation:

Use the formula $\beta _1 = r\frac{s_y}{s_x}$ $\displaystyle \beta _1 = r\frac{s_y}{s_x}$.

Plug in the given values for $\displaystyle r$ and s_x $\displaystyle s_x$ and this becomes an algebra problem.

$0.64 = 0.8\frac{s_y}{5}$ $\displaystyle 0.64 = 0.8\frac{s_y}{5}$

$0.8 = \frac{s_y}{5}$ $\displaystyle 0.8 = \frac{s_y}{5}$

$s_y = 0.8\cdot 5 = 4$ $\displaystyle s_y = 0.8\cdot 5 = 4$

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AP Statistics : How to find the least-squares regression line

Study concepts, example questions & explanations for AP Statistics

All AP Statistics Resources

Example Questions

Example Question #1 : How To Find The Least Squares Regression Line

Example Question #1 : How To Find The Least Squares Regression Line

Example Question #4 : Bivariate Data

All AP Statistics Resources

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