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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(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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Simple Linear Regression: Definition, Formula & Examples

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Chapter 8 / Lesson 2
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Simple 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.


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