Final Exam, Date Submitted: 12/15/2015
1. Discrete versus continuous variables
Which of the following variablesare best thought of as continuous, which discrete? Indicate your choice for each by checking the appropriate column.
Variable
Discrete
Continuous
(a) The number of occupied tables at Zito’s Cafe at p.m. next Friday
(b) The body temperature measurement of a participant in a liedetector test
(c) The number of students, in a class of , who improve their score from the first midterm to the second midterm
(d) The number of tickets purchased by a caller on a Rose Bowl ticket hotline





Additional Resources 
Elementary Statistics (A Brief Version), 6th Ed.
Bluman
Chapter 1: The Nature of Probability and Statistics
Section 1.2: Variables and Types of Data



Variables are classified as categorical or quantitativedepending on the values that they can take. Categorical variables are variables whose values are categories, and quantitative variables are variables whose values are numbers. Quantitative variables are further classified as discrete or continuous, and in this problem we focus on this latter classification.
A discrete variable is one whose possible values can be counted, meaning that its possible values correspond to some subset of the whole numbers. Thus, if the variable can take on only wholenumbered values, then it is discrete. The number of keystrokes used to type an email message is a discrete variable because it can take on only wholenumbered values: , , , , and so on.
A continuous variable is one whose possible values make up an interval of real numbers. For any continuous variable, the possible values include some range of numbers without any gaps. The time spent studying for the next statistics test is a continuous variable because it could take on any value in a range of numbers, with no numbers in the range excluded. The time spent studying could be, for instance, minutes, or minutes, or minutes, or any real number in between. It is not necessary that the values be whole numbers (like , , , etc.). Note also that it doesn’t matter whether time is measured in minutes or some other units. The possible values will make up an entire interval of numbers no matter the units.
Discrete variables often arise when quantities are being counted (“the number of”), and continuous variables often arise when quantities are being measured.
With these considerations in mind, we can classify the four variables in the problem:
(a) 
The number of occupied tables at Zito’s Cafe at p.m. next Friday:The number of occupied tables must be a whole number. It could be , , , , or any whole number up to the total number of tables at the cafe. It couldn’t be a number in between these numbers (like , for instance). Thus, the variable is discrete.

(b) 
The body temperature measurement of a participant in a liedetector test:The value of the temperature could be, for instance, degrees Fahrenheit, or degrees Fahrenheit, or degrees Fahrenheit, or any number in between. In fact, there are no real numbers that are excluded as possible values for the temperature in degrees Fahrenheit. (If some other temperature units are chosen, a similar comment applies.) Thus, the variable is continuous.

(c) 
The number of students, in a class of , who improve their score from the first midterm to the second midterm:The number of students who improve must be a whole number. In particular, it must be one of the numbers . (It can’t be a number between two whole numbers, like .) This means that the variable is discrete.

(d) 
The number of tickets purchased by a caller on a Rose Bowl ticket hotline:The number of tickets purchased is necessarily a whole number: , or , or , or , etc. (It is impossible to purchase, say, tickets.) This means that the variable is discrete.

The answer is:
Variable
Discrete
Continuous
(a) The number of occupied tables at Zito’s Cafe at p.m. next Friday
(b) The body temperature measurement of a participant in a liedetector test
(c) The number of students, in a class of , who improve their score from the first midterm to the second midterm
(d) The number of tickets purchased by a caller on a Rose Bowl ticket hotline



