Chapter 3: Q R3.2. (page 213)

Penguins diving A study of king penguins looked for a relationship between how deep the penguins dive to seek food and how long they stay underwater.41 For all but the shallowest dives, there is an association between x = depth (in meters) and y = dive duration (in minutes) that is different for each penguin. The study gives a scatterplot for one penguin titled “The Relation of Dive Duration (y) to Depth (x).” The scatterplot shows an association that is positive, linear, and strong.
a. Explain the meaning of the term positive association in this context.
b. Explain the meaning of the term linear association in this context.
c. Explain the meaning of the term strong association in this context.
d. Suppose the researchers reversed the variables, using x = dive duration and y = depth.

Short Answer

Expert verified

Part (a) $y -$ values tend to increase as the $x -$ values increase.

Part (b) $y -$ values tend to change by a constant amount as the $x -$ values increase by one.

Part (c) The scatterplot of the two variables shows a fairly apparent connection between the depth and the dive length.

Part (d) The correlation remains unchanged but the regression line will be changed.

Step by step solution

Part (a) Step 1: Given information

$x$ be depth in meters and $y$ be dive duration in minutes.

Part (a) Step 2: Explanation

As a result, a positive relationship suggests that the y-values tend to rise as the x-values rise. In this example, this means that as the depth of the dive increases, so does the time of the descent.

Part (b) Step 1: Explanation

As a result, a linear relationship means that as the x-values increase by one, the y-values tend to change by a constant amount. As a result, in this example, if the depth increases by one meter, the dive duration tends to change by a consistent amount.

Part (c) Step 1: Explanation

As a result, a high connection means that the points in the scatterplot stray very little from the general pattern between the variables, indicating that the scatterplot of the two variables has a very distinct pattern. As a result, in this situation, the scatterplot of the two variables shows a fairly apparent connection between the depth and the dive length.

Part (d) Step 1: Explanation

As a result, the correlation determines how strong the linear relationship between the variables is. When the two variables are swapped, the strength of the linear relationship between them is unaffected, and the correlation remains unchanged. Because the equation for the least-square regression line employs the variables $y$ and $x$ changing them will affect the regression line's equation.