In an article in Statistics and Computing [""An Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis"" (1991, pp. 119-128)], Carlin and Gelfand investigated the age (x) and length (y) of 27 captured dugongs (sea cows).

x 1.0 1.5 1.5 1.5 2.5 4.0 5.0 5.0 7.0 8.0 8.5 9.0 9.5 9.5 10.0 12.0 12.0 13.0 13.0 14.5 15.5 15.5 16.5 17.0 22.5 29.0 31.5
y 1.80 1.85 1.87 1.77 2.02 2.27 2.15 2.26 2.47 2.19 2.26 2.40 2.39 2.41 2.50 2.32 2.32 2.43 2.47 2.56 2.65 2.47 2.64 2.56 2.70 2.72 2.57"

(a) Find the least squares estimates of the slope and the intercept in the simple linear regression model. Find an estimate of σ2
(b) Estimate the mean length of dugongs at age 11.

Step-by-step explanation:

Hello!

a)

The dependent variable is

Y: length of a dugong

The explanatory variable is

X: age of a dugong

You need to estimate the linear regression of the length of the dugongs as a function of their age.

Using the given data I've estimated the regression using a statistic software:

The regression model is E(Yi)= α + βXi

The estimated model is ^Yi= a + bXi

Where a is the estimate of the intercept and b is the estimate of the slope:

a= 2.02

b= 0.03

And the estimate of the population variance of the error is Se²= 0.03

The estimated regression equation is ^Yi= 2.02 + 0.03Xi

b)

You have to estimate the length of a dugong when its age is 11 years using the model, for this all you have to do is replace X=11 in the regression line and calculate the corresponding ^Y value:

^Yi= 2.02 + 0.03*11= 2.35

The average length of an 11-year-old dugong should be 2.35.

I hope it helps!

use this formula:

p(a\b) = p(a and b)/p(b).

take it from here.