In Wanchai Computer Centers in Hong Kong, there are dozens...

In Wanchai Computer Centers in Hong Kong, there are dozens of computer shops selling multiple laptop brands. After a survey in one of them, 10 were selected. The ordered pairs show the speed of each computer's CPU in gigahertz and its price in Hong Kong dollars ( $1 \mathrm{USD}=7.78 \mathrm{HKD})$. $$ \begin{aligned} & (1.8,14,500),(1.6,12,290),(2.0,17,500),(1.6,16,500), \\ & (1.8,19,650),(2.4,21,000),(1.2,7,500),(1.4,12,500), \\ & (1.6,14,650),(2.0,18,350) \end{aligned} $$ a. Determinate the regression equation of the sample. b. Find the intercept and the slope of the equation. c. Compute the coefficient of determination and interpret its meaning in this specific context.

In Wanchai Computer Centers in Hong Kong, there are dozens of computer shops selling multiple laptop brands. After a survey in one of them, 10 were selected. The ordered pairs show the speed of each computer's CPU in gigahertz and its price in Hong Kong dollars ( $1 \mathrm{USD}=7.78 \mathrm{HKD})$.
$$
\begin{aligned}
& (1.8,14,500),(1.6,12,290),(2.0,17,500),(1.6,16,500), \\
& (1.8,19,650),(2.4,21,000),(1.2,7,500),(1.4,12,500), \\
& (1.6,14,650),(2.0,18,350)
\end{aligned}
$$
a. Determinate the regression equation of the sample.
b. Find the intercept and the slope of the equation.
c. Compute the coefficient of determination and interpret its meaning in this specific context.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model and analyze the relationship between a dependent variable and one or more independent variables. In its simplest form, simple linear regression involves fitting a straight line, known as the regression line, to the data points in such a way that the sum of the squared vertical distances (errors) between the observed values and the values predicted by the line is minimized.

Least Squares Method

The least squares method is a standard approach in regression analysis used to approximate the solution by minimizing the sum of the squares of the residuals. Residuals are the differences between the observed values and the values predicted by the regression model. This method ensures that the regression line represents the best linear approximation of the data by reducing the total error.

Regression Coefficients (Slope and Intercept)

In the context of linear regression, the slope and intercept are the regression coefficients that define the regression line. The slope represents the change in the dependent variable for each unit change in the independent variable, indicating the strength and direction of the relationship. The intercept is the value of the dependent variable when the independent variable is zero, acting as the starting point of the regression line on the y-axis.

Coefficient of Determination

The coefficient of determination, often denoted as R², measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It provides an indication of how well the regression model fits the data, with values closer to 1 indicating a better fit. In practical terms, it quantifies the explanatory power of the regression model in the specific context of the data being studied.

Transcript

00:01 All right.

00:03 So here's some data for 12 randomly selected laptop computers with their speed and their price.

00:11 So we're looking at as the speed determine price.

00:16 Because we want to develop a linear equation, a regression equation.

00:21 They can be used to describe how the price depends on the processor speed.

00:28 So let's go ahead and do that.

00:29 Well, i've actually already done it.

00:31 We've got it right here.

00:32 I've got, i used a some statistical software to calculate my coefficients for my linear equation and here they are.

00:43 This is the a value or the constant is the b value or the slope.

00:48 So the y hat would be equals negative 386 .54 plus 703 .93 .97 x where x is the speed.

01:08 And based on the equation, is there one machine that seems particularly over underpriced? so you run your data through your y hat or through your model, our model.

01:26 And i had my software run it for me.

01:29 And so here are the predicted residuals here, the predicted prices that is.

01:35 I guess i don't need these observations until it's up.

01:38 There's the predicted prices right here and the residuals.

01:47 So is there any machine that seems particularly over or underpriced? so we can look at the individual residuals.

01:56 So this one here, what is this? the computer 10, it's about $260 over the predicted price.

02:08 So you might say that's overpriced.

02:10 And then this one here, minus 278, computer two, this was almost 300 bucks under its predicted price.

02:23 So i'd say those two are some particularly under or overpriced based on this model.

02:31 In the correlation coefficient, it's a fun one to find.

02:36 I've got the corral function in my software in my spreadsheet to do that for me.

02:44 There go.

02:45 0 .83.

02:46 That's the correlation coefficient.

02:48 Pretty strong.

02:51 And positive linear relationship.

02:56 Now we haven't done yet.

02:57 Let me look at this data first...

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