[DATA] Tar and Nicotine Every year the Federal Trade Commission (FTC) must report tar and nicotine levels in cigarettes to Congress. Tar and nicotine levels of over 1200 brands of cigarettes are given to Congress and a random sample of those appear in the following table:

c. Determine sb_1.

Verified step by step guidance

1. Identify that \(s_{b_1}\) is the standard error of the slope in a simple linear regression, which measures the variability of the estimated slope \(b_1\).
2. Recall the formula for the standard error of the slope: \[s_{b_1} = \frac{s}{\sqrt{\sum (x_i - \bar{x})^2}}\] where \(s\) is the standard error of the residuals, \(x_i\) are the observed values of the independent variable (Tar), and \(\bar{x}\) is the mean of the \(x_i\) values.
3. Calculate the mean of the Tar values (\(\bar{x}\)) by summing all Tar values and dividing by the number of observations.
4. Compute the sum of squared deviations of \(x_i\) from \(\bar{x}\), i.e., calculate \(\sum (x_i - \bar{x})^2\) by subtracting the mean from each Tar value, squaring the result, and summing all these squared differences.
5. Determine the standard error of the residuals \(s\) by first calculating the residuals (differences between observed Nicotine values and predicted Nicotine values from the regression line), then computing the residual sum of squares, dividing by degrees of freedom (number of observations minus 2), and finally taking the square root. Once \(s\) and \(\sum (x_i - \bar{x})^2\) are known, plug them into the formula to find \(s_{b_1}\).

Key Concepts

Slope of the Least Squares Regression Line (b1)

The slope b1 represents the change in the response variable (y) for a one-unit increase in the explanatory variable (x). It is calculated as b1 = r(sy/sx), where r is the correlation coefficient, sy is the standard deviation of y, and sx is the standard deviation of x. This slope quantifies the strength and direction of the linear relationship between x and y.

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of data values. For variables x and y, sx and sy represent how spread out the tar and nicotine levels are, respectively. These values are essential in calculating the slope of the regression line and understanding data variability.

Correlation Coefficient (r)

The correlation coefficient r quantifies the strength and direction of the linear relationship between two variables, ranging from -1 to 1. A positive r indicates a positive association, while a negative r indicates a negative association. It is a key component in determining the slope of the regression line.