Question

The following table shows data on average per capita coffee consumption and heart disease rate in a random sample of \(10\) countries.

a. Enter the data into your calculator and make a scatter plot.

b. Use your calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot from part a.

c. Explain in words what the slope and y-intercept of the regression line tell us.

Short answer

Part a.

Part b. \(y=266.63-23.88x\)

Part c. Answer is explained in the explanation part.

Step-by-step solution

Part a. Step 1. Given information

\(10\) countries

Part a. Step 2. Explanation

Here, you just have to draw the scatter graph. Get the swim time as the \(x-\)axis and heart rate as the \(y-\)axis. Then, mark the points given in the table.

The result scatter graph as follows.

Part b. Step 1. Explanation

Mean of \(x=\bar{x}=\frac{\sum x_{i}}{n}=2.87\)

Mean of \(y=\bar{y}=\frac{\sum y_{i}}{n}=198.1\)

Then we can get \(a\) using the equation \(a=\bar{y}-\bar{bx}\)

\(a=\bar{y}-\bar{bx}=266.63\)

\(b\) can be taken using the equation \(b=\sum_{i-1}^{n}\frac{x_{i}y_{i}-n\bar{x}\bar{y}}{x^{2}_{i}-n\bar{x}^{2}}\)

\(b=-23.88\)

Then the regression linear equation can be obtained as \(y=193.88-1.495x\). When this equation plotted in the previous scatter graph, then it is as follows.

Conclusion: \(y=266.63-23.88x\)

Part c. Step 1. Explanation

The slope of a regression line (b) represents the rate of change in \(y\) as \(x\)changes. Because \(y\) is dependent \on\(x\), the slope describes the predicted values of \(y\) given \(x\). When using the ordinary least squares method, one of the most common linear regressions, slope, is found by calculating \(b\) as the covariance of \(x\) and \(y\), divided by the sum of squares (variance) of \(x\).

\(b=\sum_{i-1}^{n}\frac{x_{i}y_{i}-n\bar{xy}}{x_{i}^{2}-n\bar{x}^{2}}\)

The slope must be calculated before the \(y-\)intercept when using a linear regression, as the intercept is calculated using the slope. The slope of a regression line is used with a t-statistic to test the significance of a linear relationship between \(x\) and \(y\).

The intercept indicates the location where it intersects an axis. The slope and the intercept define the linear relationship between two variables, and can be used to estimate an average rate of change. The greater the magnitude of the slope, the steeper the line and the greater the rate of change.

When it comes to the question, we have a slope of \(-23.88\) with a \(y-\)intercept of \(266.63\). If there’s no any death from heart diseases at all, the \(y-\)intercept indicates that yearly coffee consumption in liters will be \(266.63\). While the slope has meaning, the \(y-\)intercept does not make sense.