A study of king penguins looked for a relationship between how deep the penguins dive to seek food and how long they stay underwater. For all but the shallowest dives, there is a linear relationship that is different for different penguins. The study report gives a scatterplot for one penguin titled " The relation of dive duration (DD) to depth (D)." Duration DD is measured in minutes and depth D is in meters. The report then says, " The regression equation for this bird is: DD = 2.19 + 0.0048 D.
(a) What is the slope of the regression line?.
ANSWER
minutes per meter.

(b) According to the regression line, how long does a typical dive to a depth of 250 meters last?
ANSWER
minutes.

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.

A study of king penguins looked for a relationship between how deep the penguins dive to seek food and how long they stay underwater. For all but the shallowest dives, there is a linear relationship that is different for different penguins. The study report gives a scatterplot for one penguin titled 'The relation of dive duration (DD) to depth (D).' Duration DD is measured in minutes and depth D is in meters. The report then says, 'The regression equation for this bird is: DD = 2.19 + 0.0048 D.' (a) What is the slope of the regression line? (b) According to the regression line, how long does a typical dive to a depth of 250 meters last?

Explore the relationship between dive duration and depth for king penguins using a linear regression equation: DD = 2.19 + 0.0048D. Learn how to calculate the slope, understand the meaning of the equation, and find the dive duration for a depth of 250 meters. Suitable for Grades 9-12.

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