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.

Step 1:
What is the slope of the regression line?

Step 2:
Explain in speci?c language what this slope says about this penguin's dives.

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

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

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

Step 3:
According to the regression line, how long does a typical dive to a depth of 200 meters last? Answer to 2 decimal places.

Step 4:
According to the regression line, how long does a typical dive to a depth of 210 meters last? Answer to 3 decimal places.

Step 5:
According to the regression line, how long does a typical dive to a depth of 220 meters last? Answer to 3 decimal places.

Step 6:
According to the regression line, how long does a typical dive to a depth of 230 meters last? Answer to 3 decimal places.

Step 7:
According to the regression line, how long does a typical dive to a depth of 240 meters last? Answer to 3 decimal places.

Step 8:
According to the regression line, how long does a typical dive to a depth of 150 meters last? Answer to 3 decimal places.

Step 9:
According to the regression line, how long does a typical dive to a depth of 160 meters last? Answer to 3 decimal places.

Step 10:
According to the regression line, how long does a typical dive to a depth of 170 meters last? Answer to 3 decimal places.

Step 11:
According to the regression line, how long does a typical dive to a depth of 180 meters last? Answer to 3 decimal places.

Step 12:
According to the regression line, how long does a typical dive to a depth of 190 meters last? Answer to 3 decimal places.

Step 13:
The dives varied from 40 meters to 300 meters in depth. Plot the regression line from x = 40 to x = 300. Which of the lines in the figure below is the correct regression line?

Yellow

Red

Blue

Answer 1

Concepts and reason

Regression: Regression is a technique that is used to determine relationship between two or more variables. That is, the change in the predictor variable influences the change in the dependent variable is determined. Moreover, in regression analysis which involves more than one independent variable, the change in the dependent is analyzed when the one independent variable is varied by keeping all other independent variables as constant.

If the data set is bivariate, then linear regression best suits the data. The straight line known as least squares regression line is obtained which best represents the data with two variables.

Fundamentals

The equation of the line is given by,

$\begin{array}{l}\\\hat y = {b_0} + {b_1}x\\\\{\rm{where,}}\\\\{b_0}{\rm{ - Intercept}}\\\\{b_1}{\rm{ - Regression}}\,{\rm{coefficient}}\\\\\hat y\,{\rm{ - Predicted}}\,{\rm{value}}\,{\rm{of}}\,{\rm{the}}\,{\rm{dependent}}\,{\rm{variable}}\\\\x\,{\rm{ - Independent}}\,{\rm{or}}\,{\rm{predictor}}\,{\rm{variable}}\\\end{array}$

Slope: The slope of a least squares regression line is interpreted as the predicted change in the average response variable for a one-unit change in the explanatory variable.

Intercept: The y-intercept of a regression line is interpreted as the predicted value of the response variable when the explanatory variable has a value of zero (though be wary of extrapolation in interpreting the intercept or other values outside the original data range).

(1)

From the given information, the regression equation is $DD = 2.69 + 0.0138D$ , here D represents depth of dive and DD represents the dive duration.

From the given regression equation, the value of intercept is 2.69 and the value of slope is 0.0138.

(2)

The regression equation is $DD = 2.69 + 0.0138D$ .

Here, the value of slope is 0.0138. Thus, the slope represents there is a 0.0138 minutes increase in dive duration if the depth is increased by 1 unit.

(3)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Substitute 200 for D,

$\begin{array}{c}\\DD = 2.69 + 0.0138D\\\\ = 2.69 + \left( {0.0138 \times 200} \right)\\\\ = 5.45\\\end{array}$

(4)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Substitute 210 for D,

$\begin{array}{c}\\DD = 2.69 + 0.0138D\\\\ = 2.69 + \left( {0.0138 \times 210} \right)\\\\ = 5.588\\\end{array}$

(5)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Substitute 220 for D,

$\begin{array}{c}\\DD = 2.69 + 0.0138D\\\\ = 2.69 + \left( {0.0138 \times 220} \right)\\\\ = 5.726\\\end{array}$

(6)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Substitute 230 for D,

$\begin{array}{c}\\DD = 2.69 + 0.0138D\\\\ = 2.69 + \left( {0.0138 \times 230} \right)\\\\ = 5.864\\\end{array}$

(7)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Substitute 240 for D,

$\begin{array}{c}\\DD = 2.69 + 0.0138D\\\\ = 2.69 + \left( {0.0138 \times 240} \right)\\\\ = 6.002\\\end{array}$

(8)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Substitute 150 for D,

$\begin{array}{c}\\DD = 2.69 + 0.0138D\\\\ = 2.69 + \left( {0.0138 \times 150} \right)\\\\ = 4.76\\\end{array}$

(9)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Substitute 160 for D,

$\begin{array}{c}\\DD = 2.69 + 0.0138D\\\\ = 2.69 + \left( {0.0138 \times 160} \right)\\\\ = 4.898\\\end{array}$

(10)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Substitute 170 for D,

$\begin{array}{c}\\DD = 2.69 + 0.0138D\\\\ = 2.69 + \left( {0.0138 \times 170} \right)\\\\ = 5.036\\\end{array}$

(11)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Substitute 180 for D,

$\begin{array}{c}\\DD = 2.69 + 0.0138D\\\\ = 2.69 + \left( {0.0138 \times 180} \right)\\\\ = 5.174\\\end{array}$

(12)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Substitute 190 for D,

$\begin{array}{c}\\DD = 2.69 + 0.0138D\\\\ = 2.69 + \left( {0.0138 \times 190} \right)\\\\ = 5.312\\\end{array}$

(13)

The linear regression equation is $DD = 2.69 + 0.0138D$ .

Here, the intercept is 2.69. From the given diagram, the red and blue lines are below 5 on y-axis. For 5 minutes time the depth of dive lies between 160 and 170. Hence, the blue line represents the regression line.

Ans: Part 1

The slope of the regression line is 0.0138.

Part 2

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

Part 3

The dive duration for the depth of 200 meters last is 5.45 minutes.

Part 4

The dive duration for the depth of 200 meters last is 5.588 minutes.

Part 5

The dive duration for the depth of 220 meters last is 5.726 minutes.

Part 6

The dive duration for the depth of 230 meters last is 5.864 minutes.

Part 7

The dive duration for the depth of 240 meters last is 6.002 minutes.

Part 8

The dive duration for the depth of 150 meters last is 4.76 minutes.

Part 9

The dive duration for the depth of 160 meters last is 4.898 minutes.

Part 10

The dive duration for the depth of 170 meters last is 5.036 minutes.

Part 11

The dive duration for the depth of 180 meters last is 5.174 minutes.

Part 12

The dive duration for the depth of 190 meters last is 5.312 minutes.

Part 13

The correct regression line is blue line.