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### Descriptive Statements:

- Analyze the effects of bias and sampling techniques.
- Use appropriate formats for organizing and displaying data.
- Analyze univariate and bivariate data in a variety of representations.
- Make predictions from data presented in a variety of representations.
- Analyze the use of measures of central tendency and spread.

### Sample Item:

Which of the following types of data representation would be most useful for illustrating
a correlation between two variables?

- a scatter plot
- a double bar graph
- a histogram
- a box-and-whisker plot

Correct Response and Explanation (Show Correct ResponseHide Correct Response)

**A. ** This question requires the examinee to use appropriate formats for organizing and
displaying data. The scatter plot is the only one of the four choices that displays data that occur as
ordered pairs for two variables.

### Descriptive Statements:

- Determine probabilities of simple and compound events.
- Use counting principles to calculate probabilities.
- Use a variety of visual representations to calculate probabilities.
- Demonstrate knowledge of methods for simulating probabilities.

### Sample Item:

Milkshakes come in chocolate, vanilla, and strawberry flavors. Two people ordered milkshakes, but their
orders were lost. If each person is given a randomly chosen flavor, what is the probability that they will
both get the flavor that they ordered?

^{2}/_{3}2 thirds
^{2}/_{9}2 ninths
^{1}/_{6}1 sixth
^{1}/_{9}1 ninth

Correct Response and Explanation (Show Correct ResponseHide Correct Response)

**D. ** This question requires the examinee to determine probabilities of simple and compound events.
The probability of one person getting the right flavor of shake is ^{1}/_{3}1 third.
The probability that the second person gets the right flavor is also ^{1}/_{3}1 third.
Since the two events are independent, the probability of the two people each getting the right flavor shake is
^{1}/_{3} • ^{1}/_{3} = ^{1}/_{9}1 third times 1 third equal 1 ninth.

### Descriptive Statements:

- Apply concepts of permutations and combinations to solve problems.
- Analyze sequences and series, including limits and recursive definitions.
- Use finite graphs and trees to represent problem situations.
- Apply set theory to solve problems.
- Apply principles of logic to solve problems (e.g., conditional and biconditional statements, conjunctions, negations).

### Sample Item:

Out of 600 businesses surveyed, 300 had Internet access, 450 had fax machines, and 50 had neither.
How many of the businesses surveyed had both Internet access and fax machines?

- 100
- 150
- 200
- 250

Correct Response and Explanation (Show Correct ResponseHide Correct Response)

**C. ** This question requires the examinee to apply set theory to solve problems.
Finding the number of businesses with both Internet access and fax machines means finding the number
of elements in the intersection of the sets represented by *I* and *F* as defined below.

Let *I* = the number of businesses with just Internet access.

Let *F* = the number of businesses with just fax machines.

Let *I* ∩ *F* =*I* intersection *F* = the number of businesses with both Internet access and fax machines.

Equation 1: *F* + *I* + *I* ∩ *F* = 550 (600 minus the 50 with neither)*F* plus *I* plus *I* intersection *F* equals 550 left parenthesis 600 minus the 50 with neither right parenthesis

Equation 2: *I* + *I* ∩ *F* = 300*I* plus *I* intersection *F* equals 300

Equation 3: *F* + *I* ∩ *F* = 450*F* plus *I* intersection *F* equals 450

Substituting *I* + *I* ∩ *F**I* plus *I* intersection *F*
from Equation 2 into Equation 1 gives *F* + 300 = 550*F* plus 300 equals 550, which implies
*F* = 250.*F* equals 250.

Substituting this value for *F* in Equation 3 gives 250 + *I* ∩ *F* = 450250 plus *I* intersection *F* equals 450, which implies that
*I* ∩ *F* = 200.*I* intersection *F* equals 200.