Question

Computing the regression line and making predictions

Suppose you are a dolphin trainer at SeaWorld. You teach the dolphins by rewarding them with fish treats after each successful attempt at a new trick. The following table lists the dolphins, the number of treats per success given to each, and the average number of attempts necessary for each to learn to perform the tricks.

Dolphin	Number of Treats	Number of Attempts
Diana	3	7
Frederick	2	8
Fatima	4	5
Marlin	1	8

You can use the preceding sample data to obtain the regression line, where Ŷ is the predicted value of Y:

Ŷ

=

bX + a

One formula for the slope of the regression line is as follows:

b

=

SPSSx

1. To calculate the slope, first calculate SP and SSx:

SP = ________ , and SSx =____________ .

(Hint: For SP use the computational formula and for SSx

use the definitional formula.)

2. The slope of the regression line is ________ , and the Y intercept of the regression line is ___________ .

3. The difference between Y and Ŷ for a particular sample point (observation) is called a residual. Calculate the predicted Y (Ŷ) for each of the dolphins, and then calculate the residuals.

Dolphin	Number of Treats	Number of Attempts	Predicted Y (Ŷ)	Residual
Diana	3	7
Frederick	2	8
Fatima	4	5
Marlin	1	8

On the following scatter diagram of the blue sample points (circle symbol), use the orange points (square symbol) to plot the regression line. Make sure that the orange line spans the entire graph (from left to right). A line segment will automatically connect the points.

Regression Line012345109876543210NUMBER OF ATTEMPTSNUMBER OF TREATS

4. Use the regression line to estimate the number of trials it would take to learn these tricks if a dolphin received five treats per trick. Ŷ for X = 5 would be ________ .

5. The head dolphin trainer is pressuring you to teach the dolphins many new tricks quickly. She has asked you to use the least-squares regression line to predict how fast the dolphins can learn tricks if you were to give them 10 treats. Which of the following is the most appropriate response?

a.The regression line predicts that the dolphins will need -0.5 attempts to learn a trick if they are given 10 treats.

b.The regression line was estimated using 1 to 4 treats and should not be used to predict what would happen if the dolphins were given 10 treats.

c.The regression line predicts that the dolphins will need 5.5 attempts to learn a trick if they are given 10 treats.

d.Since the regression line predicts that the dolphins will need a negative number of attempts, you can assume the dolphins need 0 attempts if given 10 treats.

Answer 1

(1)

a=(sum(y)*sum(x²)-sum(x)*sum(xy))/(n*sum(x²)-(sum(x))²)=9.5

b=(n*sum(xy)-sum(x)*sum(y))/ (n*sum(x²)-(sum(x))²)=-1

(2) slope of the line is -1 and intercept is 9.5

(3)

	x	y	y^	residual=y-y^
Diana	3	7	6.5	0.5
Frederick	2	8	7.5	0.5
Fatima	4	5	5.5	-0.5
Marlin	1	8	8.5	-0.5

(4) for x=5, y^=9.5-1*5=4.5

(5)  for x=10, y^=9.5-1*10=-0.5

b.The regression line was estimated using 1 to 4 treats and should not be used to predict what would happen if the dolphins were given 10 treats.