# Using data set E, answer the questions given below. Choose the dependent variable (the response...

## Question:

Using data set E, answer the questions given below.

DATA SET E: Microprocessor Speed (MHz) and Power Dissipation (watts) for *n* = 13 chips

Chip | Speed (MHz) | Power (watts) |
---|---|---|

1989 Intel 80486 | 20 | 3 |

1993 Pentium | 100 | 10 |

1997 Pentium II | 233 | 35 |

1998 Intel Celeron | 300 | 20 |

1999 Pentum III | 600 | 42 |

1999 AMD Athlon | 600 | 50 |

2000 Pentium 4 | 1300 | 51 |

2004 Celeron D | 2100 | 73 |

2004 Pentium 4 | 3800 | 115 |

2007 AMD Phenom | 2300 | 95 |

2008 Intel Core 2 | 3200 | 136 |

2009 Intel Core i7 | 2900 | 95 |

2009 AMD Phenom II | 3200 | 125 |

(a) What is the dependent variable (the response variable to be "explained") and the independent variable (the predictor or explanatory variable)?

(b) Obtain the regression equation.

(c) Calculate {eq}r^2 {/eq}.

## Simple Linear Regression

A simple linear regression analysis finds the line of best fit between observations of two variables. This means a simple linear regression estimate consists of a slope estimate and an intercept estimate. In other words, the {eq}m {/eq} and the {eq}b {/eq} in {eq}y= mx+ b {/eq}. You will often see linear regression estimates represented using the Greek letter {eq}\beta {/eq} (called "beta"):

$$\widehat{y} = \widehat{\beta}_0 + \widehat{\beta}_1 x $$

We call {eq}y {/eq} the dependent variable because the value of {eq}y {/eq} depends on {eq}x {/eq} (if you plug something in for {eq}x {/eq} in the equation, you get a value for {eq}y {/eq}). We call {eq}x {/eq} the independent variable because there is no limitation to what value you can plug in (whereas {eq}y {/eq} depends on {eq}x {/eq}). The hats over the betas and {eq}y {/eq} note that the values of the slope ({eq}\beta _1 {/eq}) and intercept ({eq}\beta _0 {/eq}) are estimates from our data. We need to calculate these estimates.

To do so, we need to first calculate the quantities {eq}\, \displaystyle\sum_{i = 1}^n x_i \, {/eq}, {eq}\, \displaystyle\sum_{i = 1}^n y_i \, {/eq}, {eq}\, \displaystyle\sum_{i = 1}^n x_i^2 \, {/eq}, {eq}\, \displaystyle\sum_{i = 1}^n y_i^2 \, {/eq} and {eq}\, \displaystyle\sum_{i = 1}^n x_i y_i \, {/eq}. Then we can use the following formulas to find the slope and intercept estimates:

{eq}\begin{align} \widehat{\beta _1} &= \dfrac{n\left( \displaystyle\sum_{i = 1}^n x_i y_i \right) -\left( \displaystyle\sum_{i = 1}^n x_i \right) \left( \displaystyle\sum_{i = 1}^n y_i \right)}{n\left( \displaystyle\sum_{i = 1}^n x_i^2 \right) -\left( \displaystyle\sum_{i = 1}^n x_i \right) ^2} \\ \widehat{\beta _0} &= \overline{y} - \widehat{\beta_1}\, \overline{x}. \end{align} {/eq}

The symbol {eq}\displaystyle\sum {/eq} means "add everything up", so if a variable {eq}z = \{1, 3, 5, 6\} {/eq} has four observations, then {eq}\displaystyle\sum_{i = 1}^n z_i = z_1 + z_2 + z_3 + z_4 = 1+ 3+ 5+ 6= 15 {/eq}. (You can also calculate the regression estimates using Excel and many other software packages.)

The final formula we need is for {eq}r^2 {/eq}, which is called the coefficient of determination. {eq}r^2 {/eq} is a value between 0 and 1 that tells you the proportion of variability in the dependent variable ({eq}y {/eq}) explained by the independent variable ({eq}x {/eq}). We can calculate it as follows:

{eq}r = \dfrac{n\left( \displaystyle\sum_{i = 1}^n x_i y_i \right) -\left( \displaystyle\sum_{i = 1}^n x_i \right) \left( \displaystyle\sum_{i = 1}^n y_i \right)} {\sqrt{\left[ n\left( \displaystyle\sum_{i = 1}^n x_i^2 \right) -\left( \displaystyle\sum_{i = 1}^n x_i \right) ^2\right] \, \left[ n\left( \displaystyle\sum_{i = 1}^n y_i^2 \right) -\left( \displaystyle\sum_{i = 1}^n y_i \right) ^2\right]}} {/eq}

Then, we simply square it to obtain {eq}r^2 {/eq}.

## Answer and Explanation:

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View this answer**(a)** The choice of dependent/independent variable is highly topic-specific, and it makes sense that one would want to predict the power dissipation...

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Chapter 8 / Lesson 2Simple linear regression is a great way to make observations and interpret data. In this lesson, you will learn to find the regression line of a set of data using a ruler and a graphing calculator.