The following data give X, the price charged per piece of...

The following data give $X$, the price charged per piece of plywood, and $Y$, the quantity sold (in thousands). $$ \begin{array}{cc} \hline \text { Price per Piece, } X & \text { Thousands of Pieces Sold, } Y \\ \hline \$ 6 & 80 \\ 7 & 60 \\ 8 & 70 \\ 9 & 40 \\ 10 & 0 \\ \hline \end{array} $$ a. Prepare a scatter plot of these data points. b. Compute the covariance. c. Compute and interpret $b_1$. d. Compute $b_0$ e. What quantity of plywood would you expect to sell if the price were $\$ 7$ per piece?

The following data give $X$, the price charged per piece of plywood, and $Y$, the quantity sold (in thousands).
$$
\begin{array}{cc}
\hline \text { Price per Piece, } X & \text { Thousands of Pieces Sold, } Y \\
\hline \$ 6 & 80 \\
7 & 60 \\
8 & 70 \\
9 & 40 \\
10 & 0 \\
\hline
\end{array}
$$
a. Prepare a scatter plot of these data points.
b. Compute the covariance.
c. Compute and interpret $b_1$.
d. Compute $b_0$
e. What quantity of plywood would you expect to sell if the price were $\$ 7$ per piece?

Key Concepts

Scatter Plot

A scatter plot is a type of graph used to display and assess the relationship between two quantitative variables. In the context of a bivariate dataset, each point on the plot represents an observation with its coordinates corresponding to the values of the two variables. This visualization is essential for identifying patterns, trends, or any potential outliers that might affect further analysis, such as fitting a regression line.

Covariance

Covariance is a statistical measure that indicates the extent to which two variables change together. It is calculated by averaging the product of the deviations of each pair of observations from their respective means. A positive covariance signifies that the variables tend to increase or decrease in tandem, while a negative covariance indicates that one variable tends to increase when the other decreases. This measure is foundational in understanding the relationship between variables and is a precursor to correlation analysis.

Linear Regression Slope (b1)

The slope in a simple linear regression, denoted as b1, quantifies the estimated change in the dependent variable for every one-unit change in the independent variable. It is computed from the covariance of the variables divided by the variance of the independent variable. The slope is crucial for interpreting the direction and strength of the relationship between the predictor and the outcome variables in the regression model.

Linear Regression Intercept (b0)

The intercept in a simple linear regression, denoted as b0, represents the expected value of the dependent variable when the independent variable is zero. It is computed after determining the slope and then solving for the intercept using the means of the variables. The intercept provides a baseline level for predictions from the regression model, although its practical interpretation depends on whether the value zero is within the scope of the data.

Prediction in Regression

Prediction in the context of linear regression involves using the estimated regression equation to forecast the value of the dependent variable given a certain value of the independent variable. This application is central when making informed decisions, as it allows for estimation based on observed trends captured by the relationship established through the regression parameters.

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