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In: Statistics and Probability

Do questions A-D Feeding behavior of black bream fish. In Brain and Behavior Evolution (April 2000),...

Do questions A-D

Feeding behavior of black bream fish. In Brain and Behavior Evolution (April 2000), Zoologists conducted a study of the feeding behavior of black bream fish. The zoologists recorded the number of aggressive strikes of two black bream fish feeding at the bottom of an aquarium in the 10-minute period following the addition of food. The next table lists the weekly number of strikes and age of the fish (in days).

Week ? ? ? ? ? ? ? ? ?

Number of Strikes ?? ?? ?? ?? ?? ?? ?? ?? ??

Age of Fish (days) ??? ??? ??? ??? ??? ??? ??? ??? ???

We want to fit a straight-line model relating number of strikes(y) to age of fish (x).

a. Find ???? , ???? ??? ???? .

b. Find the least squares estimators ?̂ 0 and ?̂ 1 and give the least squares prediction equation.

c. Give a practical interpretation of the value of ?̂ 1.

d. Find ? 2 , the estimator of the variance ? 2 of the random error term ?.

e. Use the least squares model to estimate the mean number of strikes if the age of fish is 170 days.

f. List the assumptions required for the regression analysis.

g. Is there evidence that age of fish contributes information for the prediction of number of strikes? Conduct a hypothesis test using ? = 0.1.

h. Find a 97 % Confidence interval for ?1.

Expert Solution

X	Y	XY	X²	Y²
120	85	10200	14400	7225
136	63	8568	18496	3969
150	34	5100	22500	1156
155	39	6045	24025	1521
162	58	9396	26244	3364
169	35	5915	28561	1225
178	57	10146	31684	3249
184	12	2208	33856	144
190	15	2850	36100	225
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
1444	398	60428	235866	22078

a)

Sample size, n = 9

x̅ = Ʃx/n = 1444/9 = 160.444444

y̅ = Ʃy/n = 398/9 = 44.2222222

SSxx = Ʃx² - (Ʃx)²/n = 235866 - (1444)²/9 = 4184.22222

SSyy = Ʃy² - (Ʃy)²/n = 22078 - (398)²/9 = 4477.55556

SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 60428 - (1444)(398)/9 = -3428.88889

b)

Slope, b1 = SSxy/SSxx = -3428.88889/4184.22222 = -0.819481

y-intercept, b0 = y̅ - b1* x̅ = 44.22222 - (-0.81948)*160.44444 = 175.70333

Regression equation :

ŷ = 175.7033 + (-0.8195) x

c)

As the value of age of fish increases by one unit than the value of Number of strikes decreases by 0.8195 units.

d)

Sum of Square error, SSE = SSyy -SSxy²/SSxx

= 4477.55556 - (-3428.88889)²/4184.22222 = 1667.64767

Estimate of variance, s² = SSE/(n-2) = 1667.64767/(9-2) = 238.2354

orchestra answered 1 year ago

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