Question

The following data give the average price received by fishermen for several species of fish in 2000 and 2010. The price is in cents per pound.

Fish	Year 2000 Price (x)	Year 2010 Price (y)
COD	13.1	56.0
FLOUNDER	15.3	166.7
HADDOCK	25.8	105.5
MENHADEN	1.8	41.3
PERCH	4.9	104.2
CHINOOK	55.4	236.8
COHO	39.3	135.6
ALBACORE	26.7	84.6
SOFT SHELLED CLAMS	47.5	222.6
LOBSTERS AMERICAN	94.7	374.7
SEA SCALLOPS	135.6	432.6
SHRIMP	47.6	225.4

1. Create a regression equation for the data.
   
   ˆy=y^= Round to 2 decimal places.
   
2. What is the correlation coefficient between the 2000 and 2010 prices.
   
   Round to 2 decimal places.
   
3. If a type of fish was 41.3 cents per pound in 2000, how much would you expect to pay for it in 2010?

Answer 1

a.

Here, the explanatory variable is 2000 prices and the response variable is 2010 prices.

Let the regression line be of the form,

y=β0+β1x+e,

where β0 is the shift from the origin orintercept and β1 is the

slope of the equation and e is the error

and e is the error

Let the ith observation be,

yi=β0+β1xi+ei, here i=1(1)12

Now, we want to minimize the error sum of square i=112ei2.

So we solve the normal equations for α and β and get,

β1=covx,ySx2 and β0=y-β1x

X=1ni=1nxi, Y=1ni=1nyi

Sxx=1n-1i(xi-x)2=sample variance of x

Syy=1n-1i(yi-y)2= sample variance of y

covx,y=1nixiyi-xy= population covariance b/w x and y.

Sxy=1n-1ixiyi-xy=sample covariance b/w x and y

All the calculations are done using excel functions.

Now, the calculations are shown below,

x_bar	42.301
y_bar	182.17
Sxx	1529.3
Syy	15084
Sxy	4569

Now, we estimate the parameters of the equation by least square method and have,

β0	55.76
β1	2.98

Therefore, the required regression equation is,

y year 2000=55.76+2.98x (year 2010)

b.

The correlation coefficient, r=cov(x,y)SxxSyy

Therefore, the correlation coefficient 45691529.3×15084 =0.9513

c.

Now, we have to find the expected price of a fish which was priced 41.3 cents per pound in 2000.

We replace x=41.3 in the regression equation and have,

The predicted price of the fish in 2010 (y) = 55.76+(2.98*41.3) = 178.834 cent per pound.