The following table contains ACT scores and GPA (grade point average) scores for 8 college students. (The ACT is an SAT-type test, popular at Midwestern schools. Instead of ranging from 200 – 800, the ACT ranges from 20 – 30.) Grade point average is based on a 4-point scale and has been rounded to one digit after the decimal.
Student GPA ACT
1 2.8 21
2 3.4 24
3 3.0 26
4 3.5 27
5 3.6 29
6 3.0 25
7 2.7 25
8 3.7 30
i) Estimate the relationship between GPA and ACT using OLS; that is, obtain the intercept and the slope estimates for the model
GPAi = α + β(ACTi) + μi
Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the GPA predicted to be if the ACT score increases by 5 points?
ii) Compute the fitted values and the residuals for each observation and verify that the residuals sum to zero.
iii) What is the predicted value of GPA when ACT = 20?

Kindly check explanation

Step-by-step explanation:

Given the data:

ACT (X) :

21

24

26

27

29

25

25

30

GPA (Y) :

2.8

3.4

3.0

3.5

3.6

3.0

2.7

3.7

Relationship between GPA and ACT using a linear regression calculator :

ŷ = 0.1022X + 0.56813

Slope = 0.1022

Intercept = 0.56813

The relationship between GPA and ACT is in the positive direction, this can be inferred from the slope value which is positive

The intercept value ; 0.56813 means with a GPA score of 0 ; predicted ACT score will be 0.56813

Explain. How much higher is the GPA predicted to be if the ACT score increases by 5 points?

If ACT score increases by 5 ; predicted GPA will be:

Increase = 0.1022(5)

ACT score will increases by 0.511

ii) Compute the fitted values and the residuals for each observation and verify that the residuals sum to zero.

y = 0.1022(21) + 0.56813 = 2.71433

y = 0.1022(24) + 0.56813 = 3.02093

y = 0.1022(26) + 0.56813 = 3.22533

y = 0.1022(27) + 0.56813 = 3.32753

y = 0.1022(29) + 0.56813 = 3.53193

y = 0.1022(25) + 0.56813 = 3.12313

y = 0.1022(25) + 0.56813 = 3.12313

y = 0.1022(30) + 0.56813 = 3.63413

Residual:

2.8 - 2.71433 = 0.08567

3.4 - 3.02093 = 0.37907

3.0 - 3.22533 = −0.22533

3.5 - 3.32753 = 0.17247

3.6 - 3.53193 = 0.06807

3.0 - 3.12313 = −0.12313

2.7 -3.12313 = −0.42313

3.7 - 3.63413 = 0.06587

0.08567+0.37907−0.22533+0.17247+0.06807−0.12313−0.42313+0.06587 = −0.00044 = 0.00

iii) What is the predicted value of GPA when ACT = 20?

ŷ = 0.1022(20) + 0.56813

= 2.61

parallel and congruent

step-by-step explanation:

the movement of translation is linear.   these have the same translation because you are using the same rule for each point of the triangle so you will see the same slope being used to get from a point on the triangle to its image.

same slope!   you should think parallel when you see those 2 word together (well unless it's the same line but it can't be because we are coming from different points here)

congruent! because each point is being moved the same distance.