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.$25$ For all but the shallowest dives, there is a linear relationship that is different for different penguins. The study gives a scatterplot for

one penguin titled “The Relation of Dive Duration $\left(y\right)$ to Depth $\left(x\right)$” Duration y is measured in minutes and depth x is in meters. The report then says, “The regression equation for this bird is: $\stackrel{⏜}{y}=2.69+0.0138x$

(a) What is the slope of the regression line? Explain in specific language what this value 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) Does the $y-$intercept of the regression line make any sense? If so, interpret it. If not, explain why not.

The regression equation for the bird is:$\stackrel{⏜}{y}=2.69+0.0138x$

A regression line is a straight line that depicts the change in a response variable$y$when an explanatory variable $x$ changes. By putting this $x$ into the equation of the line, you may use a regression line to predict the value of $y$ for any value of $x$

The equation of the least-squares regression line, $\stackrel{⏜}{y}=2.69+0.0138x$

The rate at which the anticipated response Y moves among the line as the explanatory variable X changes are represented by the slope b of the above regression line (0.0138). Specifically, when X grows by one unit, b is the projected change in Y. When the depth of the dive increases by one meter, the dive duration increases by 0.0138 minutes. Hence, the slope is: b = 0.0138

The equation of the least-squares regression line, $\stackrel{⏜}{y}=2.69+0.0138x$

When the depth is $200$ meters, we are supposed to find the dive length. That example, given X = $200$ meters, we must determine the projected value of $Y$

The following formula is used to compute the expected value of $Y$: $\stackrel{⏜}{y}=2.69+0.0138x$

$$\begin{gathered} =2.69+0.0138x200 \\ =5.45 \end{gathered}$$

Hence, this shows that length of dive for a depth of 200 meters is $5.45$ minutes.

The $Y-$intercept of the above regression line is $2.69$

When the explanatory variable $X=0$this value is the predicted response, $Y$ In the case of diving penguins, a depth of$0$ meters makes no sense.