Question

A - Suppose Wesley is a marine biologist who is interested in the relationship between the age and the size of male Dungeness crabs. Wesley collects data on 1,000 crabs and uses the data to develop the following least-squares regression line where ? is the age of the crab in months and ?ˆ is the predicted value of ? , the size of the male crab in cm.

?ˆ=8.2052+0.5693?

What is the value of ?ˆ when a male crab is 21.7865 months old? Provide your answer with precision to two decimal places.

?ˆ=

Interpret the value of ?ˆY^. The value of ?ˆY^ is:

the predicted number of crabs out of the 1,000 crabs collected that will be 21.7865 months old.

the probability that a crab will be 21.7865 months old.

the predicted size of a crab when it is 21.7865 months old.

the predicted incremental increase in size for every increase in age by 21.7865 months.

B - For a sample of students, their total SAT scores and college grade point averages (GPAs) are:

SAT	GPA
1070	3.50
1040	2.66
1120	3.54
1260	3.83
1100	2.60
1020	2.70
960	2.00

The least‑squares regression line equation of GPA on SAT score is:

?̂ =725.71+119.5?y^=725.71+119.5x

?̂ =−3.22+0.006?y^=−3.22+0.006x

?̂ =−3.65+0.008?

Answer 1

Q A

Given : -

Wesley collects data on 1,000 crabs and uses the data to develop the following least-squares regression given by

?ˆ=8.2052+0.5693?

where ? :- is the age of the crab in months

and ? :- size of the male crab in cm.

Thus Slope b = 0.5693 and intercept a = 8.2052

To find the value of ?ˆ when a male crab is 21.7865 months old i.e when ? =21.7865 .

?ˆ= 8.2052+0.5693?

    = 8.2052+0.5693 *21.7865

?ˆ= 20.60825

Thus the value of ?ˆ is 20.60825

This is nothing but predicted value of Y when ? =21.7865 .

Thus correct option is

The value of ?ˆY^ is: the predicted size of a crab when it is 21.7865 months old.

Q B

For a sample of students, their total SAT scores and college grade point averages (GPAs) are:

SAT	GPA
1070	3.50
1040	2.66
1120	3.54
1260	3.83
1100	2.60
1020	2.70
960	2.00

To find least‑squares regression line equation of GPA on SAT score

{ Note That - We will use R-Software to find least square regression if manuall calculation are required you can ask for that in comment box .}

Let   y- GPA    and x - SAT

Least square regression line is given by

= a + b * x

where

Slope b =

Intercept a = - * b

where cov(X,Y) - Covariance of x and y

            Var (X) - Variance of x

        - Mean of GPA or y ,    - Mean of SAT or x

{ Note That - We will use R-Software for calculation purpose to find least square regression , if manuall calculation are required you can ask for that in comment box }

In R , first we will import data , and then find cov(), var() , mean etc

> data.frame("SAT"=x,"GPA"=y)
   SAT GPA
1 1070 3.50
2 1040 2.66
3 1120 3.54
4 1260 3.83
5 1100 2.60
6 1020 2.70
7 960 2.00

> mean(x)              # Mean of X or SAT
[1] 1081.429

> mean(y)                 # Mean of Y or GPA
[1] 2.975714

> cov(x,y)                 # Covariance between X and Y
[1] 51.64048

> var(x)
[1] 9014.286                 # Variance of X

> b= cov(x,y)/var(x)
> b           # Estimate of slope
[1] 0.005728737

> a= mean(y) - mean(x) * b
> a           # Estimate of Intercept
[1] -3.219506

Thus a = -3.219506 -3.22

and   b = 0.005728737 0.006

So least‑squares regression line equation will be

= a + b * x

= -3.22 + 0.006 * x

Hence correct option is Option ii) ?̂ =−3.22+0.006?

The least‑squares regression line equation of GPA on SAT score is:

?̂ = −3.22 + 0.006 ?