Question

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 = 2.69 + 0.0138D.
(a) What is the slope of the regression line? (Enter your answer rounded to four decimal places.)
(b) On average, if the depth of the dive increases by one meter, what is the increase in the diving duration? (Enter your answer rounded to four decimal places.)
(c) According to the regression line, how long does a typical dive to a depth of 244 meters last? (Enter your answer rounded to three decimal places.)

Answer 1

Answer:

a) m = 0.0138

b) 0.0138 minutes

c) 6.057 minutes

Step-by-step explanation:

We are given the following in the question:

The relation of dive duration (DD) to depth (D) is given by the regression equation:

$DD = 2.69 + 0.0138D$

Duration DD is measured in minutes, and depth D is in meters.

Here, DD is the dependent variable and D is the independent variable.

Comparing the equation to a linear equation, we have,

$y = mx + c$

where m is the slope of the equation and gives the rate of change and c is the y-intercept that is value of y when x is zero.

m = 0.0138

c = 2.69

a) slope of the regression line

The slope of the regression lines, m = 0.0138

b)  increase in the diving duration,  if the depth of the dive increases by one meter

$DD(D) = 2.69 + 0.0138D\\DD(D+1) = 2.69 + 0.0138(D+1)\\\text{Subtracting the equations}\\DD(D+1)-DD(D) = 2.69 + 0.0138(D+1) - (2.69 + 0.0138D)\\DD(D+1)-DD(D) = 0.0138(D+1-D)\\DD(D+1)-DD(D) = 0.0138$

Thus, On average, if the depth of the dive increases by one meter, 0.0138 minutes is the increase in the diving duration.

c) Duration of a typical dive to a depth of 244 meters

We put D = 244

$DD = 2.69 + 0.0138(244)\\DD = 6.057\text{ minutes}$

It takes 6.057 minutes for a dive of 244 minutes.