# ISEE Upper Level Quantitative : Data Analysis

## Example Questions

### Example Question #2 : Range

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

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.

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:

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

So the mean is greater than the range.

### Example Question #1 : How To Find Range

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

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

The mode is greater than the range.

The range is greater than the mode.

The range is equal to the mode.

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:

So the range is equal to the mode.

### Example Question #4 : Range

Consider the following set of data:

Compare and .

: The sum of the median and the mean of the set

: The range of the set

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

and are equal

is greater

is greater

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:

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

So we have:

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

Therefore is greater than .

### Example Question #1 : Sets

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

15

10

12

18

10

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.

### Example Question #631 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

We can divide the natural numbers into four sets:

In which of these sets would 197 be a member?

It cannot be determined from the information given.

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 .

### Example Question #1 : Sets

Define the universal set, , as follows:

Also, let

and

Which of these sets represents the complement of  ?

Explanation:

, the union of  and   is the set of all elements in  , , or both. Merge the two sets to get:

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  are 4 and 8, so

### Example Question #3 : How To Find The Missing Part Of A List

Which of the following is an example of two sets  and  such that  ?

Explanation:

refers to the empty set, the set with no elements;  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:

and  do not have an element in common, so this is the right choice.

### Example Question #2 : How To Find The Missing Part Of A List

Define set . How could we define set  so that ?

Explanation:

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 :

then

If :

then

If :

then

If :

then

If :

then

This is the correct choice.

### Example Question #5 : How To Find The Missing Part Of A List

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)

It is impossible to tell from the information given.

(b) is greater

(a) and (b) are equal

(a) is greater

(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  ways out of 36 that this can happen, so there are  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 .

### Example Question #6 : How To Find The Missing Part Of A List

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)

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(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:

,

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 .

This makes (a) the greater quantity