Question

A researcher 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.

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 R2 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

a) Direction=1.129+.156 Velocity

b)

c) The slope estimate is .156. Hence with unit increase in angular velocity (degrees/second), the angle (in degrees) of a frog’s escape direction is increased by .156 degrees.

d) R square=.945. Thus 94.5% variation in  frog’s escape direction data is explained by the fitted regression on angular velocity.

e) Best prediction=1.129+.156*25=5.029 degrees

f) Since direction varies within 1 and 47.5 degrees and 60 degrees is far from it, it would not be possible to predict accurately through this regression line.