Astudy 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. what is the slope of the regression line?

[tex]b = 0.0138[/tex]

Step-by-step explanation:

Given

Regression Equation:

[tex]DD=2.69+0.0138D[/tex]

Required

Determine the slope of the regression line

The equation of regression is of the form

[tex]Y = a + bX[/tex]

Where b represents the slope;

Compare [tex]Y = a + bX[/tex] to the given equation

[tex]DD=2.69+0.0138D[/tex]

We have that:

[tex]Y = DD[/tex]

[tex]a = 2.69[/tex]

[tex]b = 0.0138[/tex]

[tex]X = D[/tex]

Hence; the slope is:

[tex]b = 0.0138[/tex]

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:

[tex]DD = 2.69 + 0.0138D[/tex]

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,

[tex]y = mx + c[/tex]

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

[tex]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[/tex]

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

[tex]DD = 2.69 + 0.0138(244)\\DD = 6.057\text{ minutes}[/tex]

It takes 6.057 minutes for a dive of 244 minutes.

The slope of the regression line is 0.0138

Step-by-step explanation:

The problem tells us that there is a linear relationship between the dive duration and the depth.

Linear equations have the form y = mx + b where m is the slope, y is the dependent variable and x is the independent variable.

In this problem we have that the depth is the independent variable, the dive duration is the dependent variable and the regression equation is DD = 2.69 + 0.0138D where DD is the dive duration and D is the depth. From the previous paragraph, we can observe that the slope is the coefficient of the dependent variable. Therefore, for this equation the slope is 0.0138.

(1)0.0138

(2)A. If the depth of the dive is increased by one meter, it adds 0.0138 minutes to the time spent under water.

Nos 3-12: See Explanation

Step-by-step explanation:

Given the regression equation for the relation of dive duration (DD) to depth (D).

[tex]DD = 2.69 + 0.0138D\\$Where: Duration DD is measured in minutes\\epth D is in meters.[/tex]

(1)The slope of the regression lie =0.0138

(2)

When D=1, DD = 2.69 + 0.0138(1)=2.7038

When D=2, DD = 2.69 + 0.0138(2)=2.7176

2.7176-2.7038=0.0138

Therefore, If the depth of the dive is increased by one meter, it adds 0.0138 minutes to the time spent under water.

(3) When depth, D =200 meters

DD = 2.69 + 0.0138(200)=5.45 Minutes

(4) When depth, D =210 meters

DD = 2.69 + 0.0138(210)=5.588 Minutes

(5) When depth, D =220 meters

DD = 2.69 + 0.0138(220)=5.726 Minutes

(6) When depth, D =230 meters

DD = 2.69 + 0.0138(230)=5.864 Minutes

(7) When depth, D =240 meters

DD = 2.69 + 0.0138(240)=6.002 Minutes

(8) When depth, D =150 meters

DD = 2.69 + 0.0138(150)=4.76 Minutes

(9) When depth, D =160 meters

DD = 2.69 + 0.0138(160)=4.898 Minutes

(10) When depth, D =170 meters

DD = 2.69 + 0.0138(170)=5.036 Minutes

(11) When depth, D =180 meters

DD = 2.69 + 0.0138(180)=5.174 Minutes

(12) When depth, D =190 meters

DD = 2.69 + 0.0138(190)=5.312 Minutes

The slope of the regression line is 0.01.

Step-by-step explanation:

The given regression equation for this bird is

[tex]DD=2.64+0.01D[/tex] .... (1)

where, DD is dive duration measured in minutes, and D is depth in meters.

The slope intercept form of a line is

[tex]y=mx+b[/tex] .... (2)

where, m is slope and b is y-intercept.

On comparing equation (1) and (2), we get

[tex]y=DD,x=D,m=0.01,b=2.64[/tex]

Since, m=0.01, therefore the slope of the regression line is 0.01.