# HiSET: Math : Summarizing numerical data sets

## Example Questions

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### Example Question #1 : Use Measures Of Center To Draw Inferences About Populations

Consider the following data set:

If , which is true of the mode or modes of the data set?

Possible Answers:

The set has one mode, .

The question cannot be answered from the information given.

The set has two modes, 8 and .

The set has one mode, 8.

The set has no mode.

Correct answer:

The question cannot be answered from the information given.

Explanation:

The mode of a data set is the value that occurs most frequently in the set. If two or more values tie for most frequently occurring value, then the set has multiple modes.

We show that the question of the modes of the data set cannot be answered with certainty as follows:

Suppose  assumes a value in the data set - for example, let . the data set is

,

in which the most frequently occurring value is 1, which appears . This makes 1, , the mode.

Now, suppose  assumes a value not already in the data set; for example, let . The data set becomes

,

in which two values, 2 (the ) and 8 each occur twice.

Therefore, from the information given, the mode(s) cannot be determined with any certainty.

### Example Question #1 : Summarizing Numerical Data Sets

Consider the data set

,

where  is a prime integer.

How many possible values of  make the set bimodal?

Possible Answers:

One

Four

Three

Eight

Five

Correct answer:

One

Explanation:

The mode of a data set is the value that occurs most frequently in the set. If two values tie for most frequently occurring value, then the set has two modes - it is bimodal.

The value 16 already occurs twice in the data set. For the set to be bimodal, one of the following must happen:

Case 1: , which is not possible, since 0 is not considered prime or composite.

Case 2:  must be equal to one of the other four known values, 10, 18, 21, or 25. however,  is given to be prime - that is, to have only two factors, 1 and itself. Each of 10, 18, 21, and 25 is composite, having factors not equal to 1 or itself, so  cannot assume any of these values.

Case 3:  must be equal to one of the other four known values, 10, 18, 21, or 25. Then  must be equal to half of the selected number:

Only in one case does it hold that  is a prime integer.

The correct response is one - .

### Example Question #3 : Use Measures Of Center To Draw Inferences About Populations

Consider the data set

,

where  is a prime integer.

How many possible values of  make the set bimodal?

Possible Answers:

Five

Three

Infinitely many

Two

One

Correct answer:

Two

Explanation:

The mode of a data set is the value that occurs most frequently in the set. If two values tie for most frequently occurring value, then the set has two modes - it is bimodal.

The value 29 already occurs three times in the data set. For the set to be bimodal,  must be equal to one of the values that occurs once - 17, 21, 27, 35, or 37. Since it is given that  is prime - having only two factors, 1 and itself -  can only be either of 17 and 37, the other three values having other factors.

The correct response is two.

### Example Question #4 : Use Measures Of Center To Draw Inferences About Populations

Consider the following data set:

where  is an integer from 1 to 10 inclusive.

How many possible values of  make 8 the median of the set?

Possible Answers:

One

Two

Zero

Ten

Three

Correct answer:

Three

Explanation:

The median of a set of nine data values - an odd number - is the value that appears in the middle when the values are ranked. For 8 to be the median, 8 must be in the middle - that is, four values must appear before 8, and four values must appear after 8.

In the given data set, four values - 4, 5, 5, 6 - are known to be appear before 8, so  must appear after 8. Since  is an integer from 1 to 10,  can only be 8, 9, or 10. This makes three the correct response.

### Example Question #5 : Use Measures Of Center To Draw Inferences About Populations

Consider the following data set:

where  is an integer from 1 to 10 inclusive.

How many possible values of  make 5 the median of the set?

Possible Answers:

Zero

Ten

One

Four

Six

Correct answer:

Six

Explanation:

The median of a set of eleven data values - an odd number - is the value that appears in the middle when the values are ranked. For 5 to be the median, 5 must be in the middle - that is, five values must appear before 5, and five values must appear after 8.

We can answer this question by looking at three cases.

Case 1:

Without loss of generality, assume ; this reasoning holds for any lesser value of . The data set becomes

,

and the median is 4.

Case 2:

The data set becomes

The middle value - the median - is 5.

Case 3:

Without loss of generality, assume ; this reasoning holds for any greater value of . The data set becomes

Again, the median is 5.

Therefore, we can set  equal to 5, 6, 7, 8, 9, or 10 - any of six different values - and make the median of the set 5.

### Example Question #2 : Summarizing Numerical Data Sets

Consider the following data set:

where  is an integer from 1 to 10 inclusive.

How many possible values of  make  the median of the set?

Possible Answers:

Three

Ten

Two

None

Four

Correct answer:

Three

Explanation:

The median of a set of ten data values - and even number - is the arithmetic mean of the two values that appear in the middle when the values are ranked. For  to be the median, it would have to hold that either the two middle values are both . The data set, in ascending order, would be as follows:

.

must be between 6 and 8 inclusive. Since it is given that  is an integer, there are three possibilities: 6, 7, and 8.

Three is the correct response.

### Example Question #7 : Use Measures Of Center To Draw Inferences About Populations

Consider the data set:

Which is true of the arithmetic mean , the median , and the midrange ?

Possible Answers:

Correct answer:

Explanation:

The mean of a data set is equal to the sum of the entries divided by the number of entries. There are ten entries in the set, so

The median of a data set with an even number of values is the arithmetic mean of the values of the two entries in the middle when the values are arranged in ascending order:

This value is .

The midrange of a data set is the arithmetic mean of the least and greatest elements in the set. This element is

The correct response is that

.

### Example Question #3 : Summarizing Numerical Data Sets

Janice will take twelve tests in her political science class, worth one hundred points each. Her score for the term will be the arithmetic mean of the best ten.

She has taken eleven tests already; her scores, in order, are:

74, 79, 60, 77, 54, 80, 81, 60, 66, 68, 71

How high will Janice have to score on the twelfth test in order to get a "C" in the course, which is defined to be a mean of 70 points?

Note: Assume that no extra credit is given on any test.

Possible Answers:

Janice cannot attain a score of 70 or higher.

Janice is already assured of a score of at least 70.

Correct answer:

Janice is already assured of a score of at least 70.

Explanation:

First, we test to see if she is already assured of a 70 average. The worst-case scenario for Janice is that she will score 0 points on the twelfth test,. If this happens, her grade will be the mean of the ten best tests so far. She will drop the 0 and the fifth score, 54, so her average will be the sum of the other ten tests divided by 10:

Janice is already assured of a mean of at least 70, since any score can only raise her grade.

### Example Question #9 : Use Measures Of Center To Draw Inferences About Populations

Donna will take twelve tests in her biology class, worth one hundred points each. Her score for the term will be the arithmetic mean of the best ten.

She has taken eleven tests already; her scores, in order, are:

74, 79, 60, 77, 54, 80, 81, 60, 66, 68, 71

How high will Donna have to score on the twelfth test in order to get a "B" in the course, which is defined to be a mean of 80 points?

Note: Assume that no extra credit is given on any test.

Possible Answers:

Donna cannot attain a score of 80 or higher.

Donna is already assured of a score of at least 80.

Correct answer:

Donna cannot attain a score of 80 or higher.

Explanation:

First, we test to see if she is already assured of a 80 average. The worst-case scenario for Donna is that she will score 0 points on the twelfth test,. If this happens, her grade will be the mean of the ten best tests so far. She will drop the 0 and the fifth score, 54, so her average will be the sum of the other ten tests divided by 10:

Donna has not yet attained her 80. Now, we test to determine what her twelfth test score must be. If she scores 60 or less she will drop this score as well as the 54, so we will assume that she scores more than 60. Calling this score , the mean of her scores will be the the sum of her best nine scores thus far and this unknown score , divided by 10. The mean should be greater than or equal to 80, so we can set up and solve for  in this inequality:

Note that the third (60) and fifth (54) scores have been omitted. Add the known scores to get

Multiply both sides by 10:

Subtract 656 from both sides:

Donna would have to score 144 or more on the twelfth test - an impossible feat. She cannot attain an average of 80 or greater.

### Example Question #4 : Summarizing Numerical Data Sets

Consider the following data set:

Which of the following gives the arithmetic mean of the set in terms of ?

Possible Answers:

None of the other choices gives the correct response.

Correct answer:

Explanation:

The arithmetic mean of a data set is the sum of the items in the set divided by the number of items. There are ten items, so the mean is

Simplify the numerator by combining the like terms:

Now, split the fraction, and reduce to lowest terms:

,

the correct response.

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