Let's analyze the regression equation provided for the penguin:

$\text{DD} = 2.19 + 0.0048D$

Part (a) - Slope of the regression line

The general form of a linear regression equation is:

$y = a + bx$

Where:

• $a$ is the intercept (the value of $y$ when $x = 0$).
• $b$ is the slope (the change in $y$ for a one-unit increase in $x$).

In the given equation:

• $2.19$ is the intercept.
• $0.0048$ is the slope.

The slope tells us how much the dive duration (DD) changes for each additional meter of depth (D). Thus, the slope is 0.0048 minutes per meter.

Part (b) - Dive duration for a depth of 250 meters

We are asked to find the dive duration when the penguin dives to a depth of 250 meters. Using the regression equation:

$\text{DD} = 2.19 + 0.0048 \times 250$

Now, let's calculate it:

$\text{DD} = 2.19 + 1.2 = 3.39 \, \text{minutes}$

So, for a dive to a depth of 250 meters, the typical dive duration is 3.39 minutes.

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Would you like more details on how the regression model works or any other clarifications?

Here are five questions that build on this topic:

1. How would the slope change if the penguin's dive time increased more rapidly with depth?
2. What is the significance of the intercept in the context of this problem?
3. How could we test if the relationship between dive duration and depth is truly linear for this penguin?
4. What would the predicted dive duration be at a depth of 500 meters?
5. How might the regression equation differ for another penguin?

Tip: When interpreting slope, always link the units of both variables to get a clear sense of the rate of change between them.