Week2 Matrix Notation

Matrix Notation Exercises

Matrix Notation Exercises #1

1/1 point (graded)

In R we have vectors and matrices. You can create your own vectors with the function c.

c(1,5,3,4)

They are also the output of many functions such as

rnorm(10)

You can turn vectors into matrices using functions such as rbind, cbind or matrix.

Create the matrix from the vector 1:1000 like this:

X = matrix(1:1000,100,10)

What is the entry in row 25, column 3 ?  correct 

225

$225$ 

Explanation

X[25,3]

 

You have used 1 of 5 attempts

Correct (1/1 point)

Matrix Notation Exercises #2

1/1 point (graded)

Using the function cbind, create a 10 x 5 matrix with first column x=1:10. Then columns 2*x, 3*x, 4*x and 5*x in columns 2 through 5.

What is the sum of the elements of the 7th row?  correct 

105

$$ 

Explanation

x = 1:10

X=cbind(x, 2*x, 3*x, 4*x, 5*x)

sum(X[7,])

 

You have used 1 of 5 attempts

Correct (1/1 point)

Matrix Notation Exercises #3

1/1 point (graded)

Which of the following creates a matrix with multiples of 3 in the third column?

matrix(1:60,20,3)

matrix(1:60,20,3,byrow=TRUE) correct

x=11:20;rbind(x,2*x,3*x)

x=1:40;matrix(3*x,20,2)

Explanation

You can make each of the matrices in R and examine them visually. Or you can check whether the third column has all multiples of 3 with all(X[,3]%%3==0). Note that the fourth choice does not even have a 3rd column.

PH525.2xWeek1Introduction to Linear Models

Introduction Exercises

Introduction Exercises #1

1 point possible (graded)

If you haven't done so already, install the library UsingR

install.packages("UsingR")

Then once you load it you have access to Galton's father and son heights:

library(UsingR)
data("father.son",package="UsingR")

What is the average height of the sons (don't round off)?Answer: 68.68407

Explanation

mean(father.son$sheight)

 

Introduction Exercises #2

1 point possible (graded)

One of the defining features of regression is that we stratify one variable based on others. In Statistics we use the verb "condition". For example, the linear model for son and father heights answers the question how tall do I expect a son to be if I condition on his father being x inches. The regression line answers this question for any x.

Using the father.son dataset described above, we want to know the expected height of sons if we condition on the father being 71 inches. Create a list of son heights for sons that have fathers with heights of 71 inches (round to the nearest inch).

What is the mean of the son heights for fathers that have a height of 71 inches (don't round off your answer)? (Hint: use the function round() on the fathers' heights)

 

Explanation

mean(father.son$sheight[round(father.son$fheight)==71])

or using dplyr:

library(dplyr)

filter(father.son,round(fheight)==71) %>% summarize(mean(sheight))

Introduction Exercises #3

1/1 point (graded)

We say a statistical model is a linear model when we can write it as a linear combination of parameters and known covariates plus random error terms. In the choices below, Y represents our observations, time t is our only covariate, unknown parameters are represented with letters a,b,c,d and measurment error is represented by the letter e. Note that if t is known, then any transformation of t is also known. So, for example, both Y=a+bt +e and Y=a+b f(t) +e are linear models. Which of the following can't be written as a linear model?

Y = a + bt + e

Y = a + b cos(t) + e

Y = a + b^t + e correct

Y = a + b t + c t^2 + d t^3 + e

Introduction Exercises #4

0/1 point (graded)

Supposed you model the relationship between weight and height across individuals with a linear model. You assume that the height of individuals for a fixed weight x follows a liner model Y = a + b x + e. Which of the following do you feel best describes what e represents?

Measurement error: scales are not perfect

Within individual random fluctuations: you don't weigh the same in the morning as the afternoon

Round off error introduced by the computer we use to analyze the dataincorrect

Between individual variability: people of the same height vary in their weight.correct

Explanation

Remember the model is across individuals and we fix x. People of the same height can vary greatly in other aspects of their physiology: for example different bone density or differing amounts of muscle and fat.