91Ó°ÊÓ

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. 17 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, \(D D=2.69+0.0138 D_{\text {" }}\) " a. What is the slope of the regression line? Explain in specific language what this slope says about this penguin's dives. b. According to the regression line, how long does a typical dive to a depth of 200 meters last? c. The dives varied from 40 meters to 300 meters in depth. Use the regression equation to determine \(D D\) for \(D=40\) and \(D=300\) and then plot the regression line from \(D=40\) to \(D=300\).

Step by step solution

01

Identifying the Slope

The regression equation given is: \(DD = 2.69 + 0.0138D\). The slope of the regression line is the coefficient of \(D\), which is 0.0138. This means that for every 1-meter increase in depth, the dive duration increases by approximately 0.0138 minutes.

02

Calculating Dive Duration at 200 Meters

We substitute \(D = 200\) into the regression equation: \(DD = 2.69 + 0.0138 \times 200\). Calculate the expression: \(DD = 2.69 + 2.76 = 5.45\). Thus, a typical dive to a depth of 200 meters lasts approximately 5.45 minutes.

03

Determining Dive Duration at 40 Meters

Substitute \(D = 40\) into the regression equation: \(DD = 2.69 + 0.0138 \times 40\). Calculate the expression: \(DD = 2.69 + 0.552 = 3.242\). Therefore, for a dive of 40 meters, the duration is approximately 3.24 minutes.

04

Determining Dive Duration at 300 Meters

Substitute \(D = 300\) into the regression equation: \(DD = 2.69 + 0.0138 \times 300\). Calculate the expression: \(DD = 2.69 + 4.14 = 6.83\). Thus, for a dive of 300 meters, the duration is approximately 6.83 minutes.

05

Plotting the Regression Line

To plot the regression line, use the points calculated: At \(D = 40\), \(DD = 3.24\) and at \(D = 300\), \(DD = 6.83\). Draw a straight line using these two points on a graph with the x-axis representing depth \(D\) and the y-axis representing duration \(DD\). The line represents the relationship between depth and duration for depths ranging from 40 meters to 300 meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatterplot Interpretation

When analyzing data, a scatterplot helps visualize the relationship between two variables, making it easier to identify patterns and trends. In this scenario, the scatterplot depicts the connection between the depth of king penguin dives and how long they remain submerged.
This visual representation provides immediate insights:

• Structure: Each data point on the scatterplot represents an individual dive, with the x-axis showing dive depth (in meters) and the y-axis indicating dive duration (in minutes).
• Trend Observation: By observing the scatterplot, one can discern if there is a trend or correlation. For example, if data points tend to rise together in a linear fashion, this suggests a positive correlation between depth and duration.
• Understanding Deviations: Scatterplots also reveal outliers or anomalies where dives do not follow the pattern observed in other data points.

In simpler terms, scatterplots are the first step in displaying the correlation (or lack thereof) between two quantities, paving the way for more in-depth statistical analysis.

Slope Calculation

In regression analysis, the slope of a line is a vital component, representing the rate of change between two variables. Here, the regression equation for our penguin's dives is given as: \(DD = 2.69 + 0.0138D\).The slope, in this case, is 0.0138.
This number holds the key to understanding the penguins' behavior during dives:

• Rate of Increase: The slope indicates that for every additional meter in dive depth, the duration increases by approximately 0.0138 minutes, meaning the dives last longer as penguins dive deeper.
• Behavioral Insight: Observing such a slope can provide insights into animal behavior or energy expenditure as they delve into deeper waters.
• Precision: The precise calculation of the slope allows for predicting dive duration changes with varying depths.

The consistent nature of the slope in this model suggests a steady relationship between depth and duration over the observed range.

Dive Duration Estimation

Once we have the regression equation, estimating dive duration for different depths becomes straightforward. For our king penguins, this equation predicts how long a dive will last based on its depth.
For instance, we can substitute different values to determine durations:

• For a dive depth of 200 meters: Substitute \(D = 200\) into the equation, yielding \(DD = 2.69 + 0.0138 \times 200 = 5.45\) minutes. Thus, a 200-meter dive typically lasts 5.45 minutes.
• For shallower or deeper dives, similar substitutions provide respective durations; at 40 meters, the dive might last about 3.24 minutes, and at 300 meters, it extends to 6.83 minutes.

Using regression equations allows scientists and researchers to quantify relationships like these accurately, providing predictive power for new observations.

Plotting Regression Line

Plotting the regression line is a graphical method to represent the relationship predicted by a regression equation. With the calculated dive durations at different depths, the next step involves plotting these on a graph to form a clear visual of the data.
Here’s a step-by-step guide:

• Choose Points: Use significant calculated points, such as at depths of 40 meters (duration 3.24 minutes) and 300 meters (duration 6.83 minutes).
• Graph Setup: On a graph with depth \(D\) as the horizontal x-axis and dive duration \(DD\) as the vertical y-axis, plot these points.
• Drawing the Line: Connect these points with a straight line. This line is your regression line and should reflect the equation \(DD = 2.69 + 0.0138D\), smoothly depicting the linear relationship between depth and duration from 40 to 300 meters.
• Visualization Impact: By visualizing, you can directly observe the trend and make intuitive inferences about dive duration predictions for untested depths within the range.

The regression line is not just a curve on paper; it represents understanding and predicts real-world behavioral patterns in this context.