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ISEE Upper Level Quantitative : Data Analysis and Probability

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #151 : Data Analysis And Probability

In the following set of data compare the mean and the range:

$\left \{ 4,3,8,8\right \}$

Possible Answers:

The mean and the range are equal.

It is not possible to compare the mean and the mode based on the information given

The range is greater than the mean.

The mean is greater than the range.

Correct answer:

The mean is greater than the range.

Explanation:

The mean of a set of data is given by the sum of the data, divided by the total number of values in the set. So we can write:

$Mean = \frac{4+3+8+8}{4}=\frac{23}{4}=5.75$

The range is the difference between the lowest and the highest values. So we have:

$Range = Maximum\ Value-Minimum\ Value=8-3=5$

So the mean is greater than the range.

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Example Question #152 : Data Analysis And Probability

In the following set of data compare the mode and the range:

$\left \{ 1,1,1,2,2,2,2,3,3\right \}$

Possible Answers:

The range is greater than the mode.

The range is equal to the mode.

It is not possible to compare the mean and the mode based on the information given.

The mode is greater than the range.

Correct answer:

The range is equal to the mode.

Explanation:

The mode of a set of data is the value which occurs most frequently which is in this problem.

The range is the difference between the lowest and the highest values. So we have:

$Range = Maximum\ Value-Minimum\ Value=3-1=2$

So the range is equal to the mode.

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Example Question #5 : How To Find Range

Consider the following set of data:

$\left \{ 2,4,4,5,6,10 \right \}$

Compare (a) and (b) .

(a) : The sum of the median and the mean of the set

(b) : The range of the set

Possible Answers:

(a) is greater

(b) is greater

(a) and (b) are equal

It is not possible to compare the mean and the mode based on the information given.

Correct answer:

(a) is greater

Explanation:

The mean of a set of data is given by the sum of the data, divided by the total number of values in the set. So we can write:

$Mean = \frac{2+4+4+5+6+10}{6}=\frac{31}{6}\approx 5.17$

The median is the average of the two middle values of a set of data with an even number of values. So we have:

$Median=\frac{4+5}{2}=4.5$

So we have:

$Median+Mean\approx 4.5+5.17\approx 9.67$

The range is the difference between the lowest and the highest values. So we have:

$Range = Maximum\ Value-Minimum\ Value=10-2=8$

Therefore (a) is greater than (b) .

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Example Question #1 : Sets

A class has 25 students. If 60% of them are boys, how many students are girls?

Possible Answers:

Correct answer:

Explanation:

If 60% of the students are boys, 40% are girls (100 – 60 = 40). Multiply 25 by 40% (25 * 0.4 = 10); therefore, 10 students are girls.

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Example Question #1 : How To Find The Missing Part Of A List

We can divide the natural numbers into four sets:

$A = \left \{ 1, 5, 9, 13, 17, ... \right \}$

$B= \left \{ 2, 6, 10, 14, 18...\right \}$

$C = \left \{ 3, 7, 11, 15, 19...\right \}$

$D = \left \{ 4, 8, 12, 16, 20...\right \}$

In which of these sets would 197 be a member?

Possible Answers:

It cannot be determined from the information given.

Correct answer:

Explanation:

The sets are divided according to the remainder obtained when each element is divided by 4. 197 divided by 4 yields a remainder of 1; all of the elements of match this description, so $197 \in A$ .

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Example Question #2 : Sets

Define the universal set, , as follows:

$U = \left \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\right \}$

Also, let

$A = \left \{ 1, 2, 3, 6, 9\right \}$ and $B = \left \{ 3, 5, 7, 9, 10\right \}$

Which of these sets represents the complement of $A \cup B$ ?

Possible Answers:

$\left \{ 1, 2, 5, 6, 7, 10\right \}$

$\left \{ 1, 2, 3, 5, 6, 7, 9, 10\right \}$

$\left \{ 3,9\right \}$

$\left \{ 3, 4, 8 ,9\right \}$

$\left \{ 4, 8\right \}$

Correct answer:

$\left \{ 4, 8\right \}$

Explanation:

$A \cup B$ , the union of and is the set of all elements in , , or both. Merge the two sets to get:

$A \cup B = \left \{ 1, 2, 3, 5, 6, 7, 9, 10\right \}$

The complement of a set is the set of all elements in the universal set that are not in the set. The only elements in not in $A \cup B$ are 4 and 8, so

$\overline{ \left (A \cup B \right )}= \left \{ 4,8\right \}$

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Example Question #1 : How To Find The Missing Part Of A List

Which of the following is an example of two sets and such that $A \cap B = \varnothing$ ?

Possible Answers:

$A = \left \{ \textrm{John, Mary, Fred, Stan, Harry } \right \}$

$B = \left \{ \textrm{Greg, Larry, Phil, Andy, Joanne }\right \}$

$A = \left \{ 2, 8, 4, 1, 10\right \}$

$B = \left \{ 5, 9, 7, 2, 0\right \}$

$A = \left \{ \Delta , \Gamma , \Theta , \Lambda , \Sigma \right \}$

$B = \left \{ \Omega , \Psi , \Phi , \Pi , \Lambda \right \}$

$A = \left \{ a,v,r, w,s,y\right \}$

$B = \left \{ k, p, r, q,x\right \}$

$A = \left \{ \textrm{Lincoln, Washington, Jefferson, Jackson, Madison}\right \}$

$B = \left \{ \textrm{ Monroe, Kennedy, Madison, McKinley , Taft} \right \}$

Correct answer:

$A = \left \{ \textrm{John, Mary, Fred, Stan, Harry } \right \}$

$B = \left \{ \textrm{Greg, Larry, Phil, Andy, Joanne }\right \}$

Explanation:

$\varnothing$ refers to the empty set, the set with no elements; $A \cap B = \varnothing$ if and only if the two sets have no elements in common. In four of these cases, and share an element, which in each of these four choices is underlined:

$A = \left \{ \Delta , \Gamma , \Theta , \underline{\Lambda} , \Sigma \right \}$

$B = \left \{ \Omega , \Psi , \Phi , \Pi , \underline{\Lambda} \right \}$

$A = \left \{ \textrm{Lincoln, Washington, Jefferson, Jackson, \underline{Madison}}\right \}$

$B = \left \{ \textrm{ Monroe, Kennedy, \underline{Madison}, McKinley , Taft} \right \}$

$A = \left \{ a,v,\underline{r}, w,s,y\right \}$

$B = \left \{ k, p, \underline{r}, q,x\right \}$

$A = \left \{ \underline{2}, 8, 4, 1, 10\right \}$

$B = \left \{ 5, 9, 7, \underline{2}, 0\right \}$

$A = \left \{ \textrm{John, Mary, Fred, Stan, Harry } \right \}$ and $B = \left \{ \textrm{Greg, Larry, Phil, Andy, Joanne }\right \}$ do not have an element in common, so this is the right choice.

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Example Question #4 : Sets

Define set $A = \left \{ 1, 3, 5, 6, 7, 8, 10 \right \}$ . How could we define set so that $A \cap B = \left \{ 1, 5, 6, 8\right \}$ ?

Possible Answers:

$B = \left \{ 1, 2, 5, 6, 8, 9, 11, 12\right \}$

$B = \left \{ 1, 2, 5, 6, 9, 12, 13, 15\right \}$

$B = \left \{5, 6,8, 9, 16\right \}$

$B = \left \{ 1,5, 10\right \}$

$B = \left \{ 1, 2, 4, 5, 6, 10, 11, 12\right \}$

Correct answer:

$B = \left \{ 1, 2, 5, 6, 8, 9, 11, 12\right \}$

Explanation:

$A \cap B$ is the set of all elements in both and .

We can test each set and determine which elements are shared by that set and :

If $B = \left \{ 1, 2, 5, 6, 9, 12, 13, 15\right \}$ :

then

$A \cap B$

$= \left \{ \underline{1}, 3, \underline{5}, \underline{6}, 7, 8, 10 \right \} \cap \left \{ \underline{1}, 2,\underline{ 5}, \underline{6}, 9, 12, 13, 15\right \}$

$= \left \{ 1,5,6\right \}$

If $B = \left \{ 1, 2, 4, 5, 6, 10, 11, 12\right \}$ :

then

$A \cap B$

$= \left \{ \underline{1}, 3, \underline{5}, \underline{6}, 7, 8, \underline{10} \right \} \cap \left \{ \underline{1}, 2, 4, \underline{5}, \underline{6}, \underline{10}, 11, 12\right \}$

$= \left \{ 1,5,6, 10\right \}$

If $B = \left \{ 1,5, 10\right \}$ :

then

$A \cap B$

$= \left \{ \underline{1}, 3, \underline{5},6, 7, 8, \underline{10} \right \} \cap \left \{ \underline{1}, \underline{5}, \underline{10}\right \}$

$= \left \{ 1,5, 10\right \}$

If $B = \left \{5, 6,8, 9, 16\right \}$ :

then

$A \cap B$

$= \left \{ 1, 3, \underline{5}, \underline{6}, 7, \underline{8}, 10 \right \} \cap\left \{\underline{5}, \underline{6},\underline{8},9, 16\right \}$

$= \left \{ 5,6,8\right \}$

If $B = \left \{ 1, 2, 5, 6, 8, 9, 11, 12\right \}$ :

then

$A \cap B$

$= \left \{ \underline{1}, 3, \underline{5}, \underline{6}, 7, \underline{8}, 10 \right \} \cap \left \{ \underline{1}, 2, \underline{5}, \underline{6}, \underline{8}, 9, 11, 12\right \}$

$= \left \{ 1,5,6, 8\right \}$

This is the correct choice.

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Example Question #153 : Data Analysis And Probability

A pair of fair dice are rolled. Which is the greater quantity?

(a) The probability that at least one die comes up 5 or 6

(b) $\frac{1}{2}$

Possible Answers:

(b) is greater

(a) is greater

It is impossible to tell from the information given.

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

For the roll to be unfavorable to the event that at least one of the dice is 5 or 6, both dice would have to be 1, 2, 3, or 4. There are $4 \times 4 = 16$ ways out of 36 that this can happen, so there are 36-16 = 20 ways for either or both of the two dice to be 5 or 6. Since half of 36 is 18, the probability of this event is greater than $\frac{1}{2}$ .

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Example Question #5 : Sets

A pair of fair dice are tossed. Which is the greater quantity?

(a) The probability that the product of the two dice will be an even number

(b) $\frac{1}{2}$

Possible Answers:

(a) is greater

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

The product of the two dice will be an odd number only if both dice are odd; the rolls favorable to that event are:

$\left \{ (1,1) , (1,3), (1,5), (3,1) , (3,3), (3,5),(5,1) , (5,3), (5,5) \right \}$ ,

or nine out of the thirty-six possible rolls. This makes twenty-seven of the thirty-six equally probable rolls favorable to getting an even product. Since half of 36 is 18, the probability of getting an even product is greater than $\frac{1}{2}$ .

This makes (a) the greater quantity

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Data Analysis and Probability

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #151 : Data Analysis And Probability

Example Question #152 : Data Analysis And Probability

Example Question #5 : How To Find Range

Example Question #1 : Sets

Example Question #1 : How To Find The Missing Part Of A List

Example Question #2 : Sets

Example Question #1 : How To Find The Missing Part Of A List

Example Question #4 : Sets

Example Question #153 : Data Analysis And Probability

Example Question #5 : Sets

All ISEE Upper Level Quantitative Resources

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