Question

6. A biologist wishes to examine the collision avoidance behaviour of bullfrogs housed in experimental boxes. He placed an object in the upper visual field of each frog and then moved the object progressively closer until the frog jumped to avoid collision. He is interested in how the angle (in degrees) of a frog’s escape direction depends on its angular velocity (degrees/second). The data (for only the 69 frogs whose escape direction was less than 50 degrees) are available in the file frog50.csv.

"Direction","Velocity"

1.9,22.3

4.9,34.8

6.6,57.3

16.9,124.6

5.8,14.1

32,191.6

26,129.3

1.1,8.5

6.7,33.1

1.7,1.7

19.1,125.3

9.5,35.9

9.2,50.9

35.2,183.2

7.3,53.2

47.5,286.7

37.5,173.4

4.8,19.8

21.4,90.3

1.4,7.2

1.9,7.5

2.4,10.6

1,17.5

1.3,6.1

11.7,83.8

24.3,193.2

1.4,27

1.6,11.5

9.7,59.2

44.4,265.8

4.6,5.1

4.2,31

1.8,17.3

21.3,154.6

19.1,88.5

4.9,10

15.6,96.6

30,178

38.1,299.6

1.2,7.1

12.5,70.4

43.7,269.8

6.7,9.1

1.1,0.6

32,237.2

8.1,35.9

16.6,74.3

25.1,130.3

2.7,26

1.7,17.4

6.9,39.2

8.4,17.2

2,3.7

47.5,280.2

2.9,0.1

1.2,8.8

15.4,97.4

19.8,104.5

3.8,12.4

16.1,91

1.5,8

6.3,51.7

25,157.1

3,10

36.6,193.8

5.3,21.4

4.5,13.5

3.8,43.4

3.1,27.8

a. Give the equation of the fitted regression line.

b. Create a plot of direction vs. velocity, including the fitted regression line.

c. Interpret the slope of the fitted regression line in the language of the problem.

d. State the R 2 value and its interpretation (in the language of the problem).

e. What is the best prediction of the velocity of a jump with a direction of 25 degrees?

f. Can you provide an accurate prediction of the velocity of a jump with a direction of 60 degrees? If so, provide this prediction. If not, explain why.

Answer 1

Upon regressing the above data,we get the following results..

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.971868
R Square	0.944527
Adjusted R Square	0.943699
Standard Error	19.74085
Observations	69
ANOVA
	df	SS	MS	F	Significance F
Regression	1	444569.8	444569.8	1140.797	8.43E-44
Residual	67	26109.98	389.7013
Total	68	470679.8
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-2.59398	3.332464	-0.7784	0.439075	-9.24561	4.05764	-9.24561	4.05764
Direction	6.047301	0.179043	33.77568	8.43E-44	5.689929	6.404672	5.689929	6.404672

A. The regression line equation is given as

Y(Velocity) = -2.59398 + 6.0473* X( Direction in angles)

B. The scatter plot between velocity and direction is shown below:

C. In this case, the slope of Direction is 6.0473, the p value with respect to the slope is 8.43 * 10^-44. We can infer the following inferences:

1. The slope is positive, which means Velocity increases with Direction. This is validated from the scatter plot, where both Direction and Velocity move in same direction.

2. The slope is 6.0473, which means for an increase in Direction by 1 degree, the velocity increases by 6.0473 degrees/ second.

3. Here , the p value is 8.43 * 10^-44, which is less than 0.05. So ,we can conclude that the slope is significant.

D. The R² in this case is 0.9445 or 94.45%. It means that 94.45% of the variation in velocity is explained by the variation in the Direction.