Questions: The following table shows the number of newspaper subscriptions in Middletown, USA where t represents the number of years since 2002 (t=0 in 2002) and S represents the total subscriptions each year-measured in thousands. t 0 2 4 6 8 S 411 392 203 138 31 Use your graphing calculator to determine the regression equation in the form of S=at+b. What is the slope of the regression equation?

Solution Steps

The means of the independent variable \( t \) and the dependent variable \( S \) are calculated as follows:

\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 4.0 \]

\[ \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 235.0 \]

The correlation coefficient \( r \) is found to be:

\[ r = -0.9772 \]

The numerator for the slope \( \beta \) is calculated as:

\[ \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 2672 - 5 \cdot 4.0 \cdot 235.0 = -2028.0 \]

The denominator for the slope \( \beta \) is:

\[ \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 120 - 5 \cdot 4.0^2 = 40.0 \]

Thus, the slope \( \beta \) is:

\[ \beta = \frac{-2028.0}{40.0} = -50.7 \]

The intercept \( \alpha \) is calculated using the formula:

\[ \alpha = \bar{y} - \beta \bar{x} = 235.0 - (-50.7) \cdot 4.0 = 437.8 \]

The line of best fit can be expressed as:

\[ S = 437.8 - 50.7t \]

Final Answer