Homework 3: The normal distribution
4 book questions, some R commands
Due 2/23
This homework begins with reading Sections 3.6, 4.1-4.4 and 4.6 of your textbook.
(This will help with your lab due next week also.) The text spends a fair bit of space
in Section 4.3 explaining how to look up information on a normal table. This is not a
skill we will emphasize in this class. Instead of understanding the mechanics of table
lookup, we would prefer that you understand the relationship between a standard
normal and a normal with any other mean µand standard deviation σ.
Questions: 4.3, 4.4, 4.16, 4.18
Rather than use a table to compute the probabilities the book asks for, we would
prefer you use the functionality in R. Specifically, R provides four functions to
help you with the normal distribution: pnorm,qnorm,dnorm and rnorm. (The “p”
because it returns probabilities; the “q” because it returns quantiles – or percentiles
as your book calls them; the “d” because it returns the bell curve or density; and
the “r” because it generates random samples.)
First, suppose we are asked about a normal distribution with with mean µ= 5 and
standard deviation σ= 2. We can find the area under the associated bell curve to
the left of 6 (that is, find the chance that Y≤6 if the distribution of Yhas a bell
shape centered at µ= 5 with σ= 2) with the command
pnorm(6,m=5,s=2)
If you leave out the arguments mand s, then R will assume you are referring to the
standard normal distribution with mean 0 and standard deviation 1. As you will
see in the book, you can transform the problem above to one involving a standard
normal so that the command
pnorm((6-5)/2)
should give you the same answer. Before computers were so commonplace (or to
help instructors give in-class exams), a lot of emphasis was placed on transforming
the problem to so-called standard units and looking things up in the table. Since
