Find the equation of the regression line for the data. Then...

Find the equation of the regression line for the data. Then construct a scatter plot of the data and draw the regression line. (Each pair of variables has a significant correlation.) Then use the regression equation to predict the value of $y$ for each of the $x$ -values, if meaningful. If the $x$ -value is not meaningful to predict the value of $y,$ explain why not. If convenient, use technology. Height and Number of Stories The heights (in feet) and the numbers of stories of nine notable buildings in Atlanta (Source: Emporis Corporation) $$\begin{array}{|lc|c|c|c|c|c|c|c|c|} \hline \text { Height, } \boldsymbol{x} & 869 & 820 & 771 & 696 & 692 & 676 & 656 & 492 & 486 \\ \hline \text { Stories, } \boldsymbol{y} & 60 & 50 & 50 & 52 & 40 & 47 & 41 & 39 & 26 \\ \hline \end{array}$$ (a) $x=800$ feet (b) $x=750$ feet (c) $x=400$ feet (d) $x=625$ feet

Find the equation of the regression line for the data. Then construct a scatter plot of the data and draw the regression line. (Each pair of variables has a significant correlation.) Then use the regression equation to predict the value of $y$ for each of the $x$ -values, if meaningful. If the $x$ -value is not meaningful to predict the value of $y,$ explain why not. If convenient, use technology.
Height and Number of Stories The heights (in feet) and the numbers of stories of nine notable buildings in Atlanta (Source: Emporis Corporation)
$$\begin{array}{|lc|c|c|c|c|c|c|c|c|}
\hline \text { Height, } \boldsymbol{x} & 869 & 820 & 771 & 696 & 692 & 676 & 656 & 492 & 486 \\
\hline \text { Stories, } \boldsymbol{y} & 60 & 50 & 50 & 52 & 40 & 47 & 41 & 39 & 26 \\
\hline
\end{array}$$
(a) $x=800$ feet
(b) $x=750$ feet
(c) $x=400$ feet
(d) $x=625$ feet

Key Concepts

Regression Analysis

Regression analysis is a statistical method used to model and analyze the relationship between two or more variables. In the context of a linear regression, it involves finding the best-fitting straight line through a set of data points using techniques such as the least squares method.

Scatter Plot

A scatter plot is a visual tool that displays individual data points on a graph, with one variable on the horizontal axis and another on the vertical axis. It is used to visually assess the relationship or correlation between the two variables.

Least Squares Method

The least squares method is a formal approach used in linear regression to determine the regression line that minimizes the sum of the squared differences between the observed values and the values predicted by the line. This method ensures the best overall fit of the line to the data.

Prediction and Extrapolation

Once a regression line is established, it can be used to predict the value of the dependent variable for a given independent variable by substituting into the regression equation. However, caution must be exercised when making predictions for values outside the range of the observed data (extrapolation), as these predictions may not be reliable.

Correlation and Data Domain

Correlation refers to the degree to which two variables are linearly related. When dealing with significant correlations, it is important to consider the domain of the original data; predictions made within the range of the observed values are typically more meaningful, whereas values outside this range may lead to unreliable forecasts.

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