Data Analysis, Probability, and Statistics - HiSET
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Researchers performed a survey of the populations of several small islands. The data is represented by the following histogram.
What percentage of islands surveyed had a population of less than 140?
Researchers performed a survey of the populations of several small islands. The data is represented by the following histogram.
What percentage of islands surveyed had a population of less than 140?
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On the horizontal axis of a histogram, you have labels representing ranges of values. For example, the label "140–159" represents the range of islands with populations between 140 and 159.
The question asks for the percent of islands with populations under 140. The ranges that are smaller than 140 are the "120–139" range and the "100–119" range. Match the bars of each with the percents on the vertical (y) axis. The percents corresponding to these ranges are 35% and 30%, respectively.
Adding both of these percents yields the answer, 65%.
On the horizontal axis of a histogram, you have labels representing ranges of values. For example, the label "140–159" represents the range of islands with populations between 140 and 159.
The question asks for the percent of islands with populations under 140. The ranges that are smaller than 140 are the "120–139" range and the "100–119" range. Match the bars of each with the percents on the vertical (y) axis. The percents corresponding to these ranges are 35% and 30%, respectively.
Adding both of these percents yields the answer, 65%.
Determine the linear regression line for the following paired data:
Determine the linear regression line for the following paired data:
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The formula for a regression line for a set of paired data is the line of the equation
,
where
and
.
, the number of pairs.
is the sum of the 
values:
is the sum of the 
values:
is the sum of the squares of the 
values:
is the sum of the 
products:
Substituting:
and 
are the arithmetic means of the 
and 
values, respectively - the sums divided by the number of values:
The equation of the linear regression line is 
.
The formula for a regression line for a set of paired data is the line of the equation
where
and
Substituting:
The equation of the linear regression line is 
Determine the linear regression line for the following paired data:
Determine the linear regression line for the following paired data:
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The formula for a regression line for a set of paired data is the line of the equation
,
where
and
.
, the number of pairs.
is the sum of the 
values:
is the sum of the 
values:
is the sum of the squares of the 
values:
is the sum of the 
products:
Substituting:
and 
are the arithmetic means of the 
and 
values, respectively - the sums divided by the number of values:
The linear regression line is the line of the equation 
.
The formula for a regression line for a set of paired data is the line of the equation
where
and
Substituting:
The linear regression line is the line of the equation 
The yearly revenue over 10 years of a company is given by the following graph.
Using the graph, determine the approximate revenue in the year 2004.
The yearly revenue over 10 years of a company is given by the following graph.
Using the graph, determine the approximate revenue in the year 2004.
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The question asks for you to find the revenue in the year 2004. The horizontal axis is labeled "Year," so look there first to find where the label for 2004 is.
Tracing upwards, you will see that the value 2004 corresponds to a value of about 16 on the vertical axis. Since the label for the vertical axis is "Dollars in Millions," this means that the revenue in 2004 was 16 million dollars.
In numerical form, 16 million dollars is $16,000,000.
The question asks for you to find the revenue in the year 2004. The horizontal axis is labeled "Year," so look there first to find where the label for 2004 is.
Tracing upwards, you will see that the value 2004 corresponds to a value of about 16 on the vertical axis. Since the label for the vertical axis is "Dollars in Millions," this means that the revenue in 2004 was 16 million dollars.
In numerical form, 16 million dollars is $16,000,000.
The approximate salaries of employees in a company are given by the histogram below:
Using the data presented in the histogram, determine what percentage of employees earn a weekly salary of $180,000 or more.
The approximate salaries of employees in a company are given by the histogram below:
Using the data presented in the histogram, determine what percentage of employees earn a weekly salary of $180,000 or more.
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Each bar in this histogram represent the percent of employees with salary in the given range. For example, the bar above the label "150–159" means that 25% of the employees earn between $150,000 and $159,000.
Since we want the percent of employees who earn $180,000 or more, we need to add the values for the bars for "180–189", "190–199", and "200–210".
The bars show that 15% of employees earn between $180,000 and $189,000, 15% earn between $190,000 and $199,000, and 5% earn between $200,000 and $210,000.
Adding these percentages together yields 35%.
Each bar in this histogram represent the percent of employees with salary in the given range. For example, the bar above the label "150–159" means that 25% of the employees earn between $150,000 and $159,000.
Since we want the percent of employees who earn $180,000 or more, we need to add the values for the bars for "180–189", "190–199", and "200–210".
The bars show that 15% of employees earn between $180,000 and $189,000, 15% earn between $190,000 and $199,000, and 5% earn between $200,000 and $210,000.
Adding these percentages together yields 35%.
Which of these data sets would be best represented using a line graph?
Which of these data sets would be best represented using a line graph?
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The line graph is best used with data that occurs sequentially, especially sequentially over a period of time. The line graph is best at describing a general trend, such as an increase or decrease, over time.
The decrease in weight of an adult male over 12 months fits a line graph well because there is data that can be organized sequentially (chronologically, in this case) and because we would be looking for some sort of trend in the data, a decrease over time in this case.
The line graph is best used with data that occurs sequentially, especially sequentially over a period of time. The line graph is best at describing a general trend, such as an increase or decrease, over time.
The decrease in weight of an adult male over 12 months fits a line graph well because there is data that can be organized sequentially (chronologically, in this case) and because we would be looking for some sort of trend in the data, a decrease over time in this case.
The revenue of a small company over 10 years is portrayed by the following line graph.
Which of the following statements can best be made about the data?
The revenue of a small company over 10 years is portrayed by the following line graph.
Which of the following statements can best be made about the data?
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The question asks you to identify a trend that best fits the data.
Over 10 years, the revenue of the company decreases and increases with no clear overall pattern or trend, so the answers claiming that there is a general increase or decrease are incorrect.
While the revenue was at a high point ($65,000) in the year 1994, it was not the highest point in the 10 years. The highest point was in 1998, with a revenue of $77,000.
While the revenue did increase from 1991 to 1992, it decreased by even more from 1991 to 1993. Therefore, from 1991 to 1993, there was actually a net decrease in revenue, not a net increase.
The correct answer is: "The company's revenue generally decreased from 1994 to 1996." Notice the data points matching the years 1994, 1995, and 1996 are linked by a line that has a downward slope. This indicates a general decrease in revenue.
The question asks you to identify a trend that best fits the data.
Over 10 years, the revenue of the company decreases and increases with no clear overall pattern or trend, so the answers claiming that there is a general increase or decrease are incorrect.
While the revenue was at a high point ($65,000) in the year 1994, it was not the highest point in the 10 years. The highest point was in 1998, with a revenue of $77,000.
While the revenue did increase from 1991 to 1992, it decreased by even more from 1991 to 1993. Therefore, from 1991 to 1993, there was actually a net decrease in revenue, not a net increase.
The correct answer is: "The company's revenue generally decreased from 1994 to 1996." Notice the data points matching the years 1994, 1995, and 1996 are linked by a line that has a downward slope. This indicates a general decrease in revenue.
An individual's income over 10 years is represented by the following line chart.
Approximate individual's highest annual income from 1994 to 1998.
An individual's income over 10 years is represented by the following line chart.
Approximate individual's highest annual income from 1994 to 1998.
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Notice that the question narrows the time period you are examining to 1994–1998. While there are years with higher incomes than in this time period (namely, 1992 and 1999), we can only consider the years from 1994 to 1998.
The highest point within this period is in the year 1998. Matching this data point with the values on the y-axis (the income values), you will see that this data point lies between $400,000 and $500,000. Looking more closely, you can even say that it is above halfway between the values, so it appears to be greater than $450,000.
Thus, the value that approximates the income in 1998 is $460,000.
Be careful to take into account the units on each axis. The y-axis measures income in thousands of dollars, so the answer is $460,000 and not $460.
Notice that the question narrows the time period you are examining to 1994–1998. While there are years with higher incomes than in this time period (namely, 1992 and 1999), we can only consider the years from 1994 to 1998.
The highest point within this period is in the year 1998. Matching this data point with the values on the y-axis (the income values), you will see that this data point lies between $400,000 and $500,000. Looking more closely, you can even say that it is above halfway between the values, so it appears to be greater than $450,000.
Thus, the value that approximates the income in 1998 is $460,000.
Be careful to take into account the units on each axis. The y-axis measures income in thousands of dollars, so the answer is $460,000 and not $460.
If I have 
quarters, what is the probability that I will get no more than 
heads?
If I have 
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Step 1: If we have 3 quarters and each quarter has two outcomes, then there are 
total outcomes that can come out of all of the rolls of the quarters.
Those rolls are:
HHH, HHT, HTH, THH, THT, TTH, TTT, HTT.
Step 2: Determine how many outcomes have no more than 
heads....
We cannot count HHH because it has 
heads.
We also cannot count TTT because it has no heads at all.
Step 3: We had 
options and we can't take 
options..so we have a total of 
options left.
Step 1: If we have 3 quarters and each quarter has two outcomes, then there are 
Those rolls are:
HHH, HHT, HTH, THH, THT, TTH, TTT, HTT.
Step 2: Determine how many outcomes have no more than 
We cannot count HHH because it has 
We also cannot count TTT because it has no heads at all.
Step 3: We had 
In a sack, I have 20 red marbles, 15 blue marbles and 5 green marbles. What is the chance, without replacement, that I pick a green marble first and then pick a blue marble?
In a sack, I have 20 red marbles, 15 blue marbles and 5 green marbles. What is the chance, without replacement, that I pick a green marble first and then pick a blue marble?
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Step 1: Find the probability of getting a green marble...
The probability of a green marble is 
.
Step 2: Find the probability of getting a blue marble (without replacement)
When we have without replacement, we do not put the first marble we take out back into the sack. In this case, the total number of marbles in the sack is 
less.
Probability of a blue marble is 
.
Step 3: To find the probability of both events happening at the same time, we need to multiply the probabilities in the first two steps.
So, 
Step 4: Simplify...
is the answer.
Step 1: Find the probability of getting a green marble...
The probability of a green marble is 
Step 2: Find the probability of getting a blue marble (without replacement)
When we have without replacement, we do not put the first marble we take out back into the sack. In this case, the total number of marbles in the sack is 
Probability of a blue marble is 
Step 3: To find the probability of both events happening at the same time, we need to multiply the probabilities in the first two steps.
So,
Step 4: Simplify...
What is the probability that we will roll a one on a six sided die and flip a coin to reveal "heads"?
What is the probability that we will roll a one on a six sided die and flip a coin to reveal "heads"?
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In order to solve this standard we need to understand two primary components: probabilities and the property of independent events. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:
An intersection is the likelihood that one event will occur with another.
We can calculate that the probability of rolling a one on a six-sided die is:
Now, let's determine the probability of flipping a coin to reveal the "heads" side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:
An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.
Convert to a percentage.
In order to solve this standard we need to understand two primary components: probabilities and the property of independent events. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:
An intersection is the likelihood that one event will occur with another.
We can calculate that the probability of rolling a one on a six-sided die is:
Now, let's determine the probability of flipping a coin to reveal the "heads" side. A coin has two sides: heads or tails. The probability of rolling heads is as follows:
An intersection is the probability of two events occurring simultaneously; therefore, we need to multiply the probabilities.
Convert to a percentage.
The spades and the aces are removed from a standard deck of 52. Give the probability that a card drawn at random will be a red three.
The spades and the aces are removed from a standard deck of 52. Give the probability that a card drawn at random will be a red three.
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In a deck of 52 cards, one-fourth of the cards, or 13 cards, are spades. Also, there are four aces, one of which is a spade and three of which are not spades. It follows that the removal of the spades and the aces results in the removal of
cards, to leave a modified deck of
cards.
None of the two red threes (the three of clubs, the three of diamonds) were removed, so the probability of a random draw of a red three is
.
In a deck of 52 cards, one-fourth of the cards, or 13 cards, are spades. Also, there are four aces, one of which is a spade and three of which are not spades. It follows that the removal of the spades and the aces results in the removal of
cards, to leave a modified deck of
cards.
None of the two red threes (the three of clubs, the three of diamonds) were removed, so the probability of a random draw of a red three is
How many ways can eight people be seated at two tables if four people should be at each table, and it only matters at which table each person sits (ass opposed to which seat)?
How many ways can eight people be seated at two tables if four people should be at each table, and it only matters at which table each person sits (ass opposed to which seat)?
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The number of ways to choose four people to sit at the first table without regard to the arrangement at either table is the number of combinations of four people from a set of eight; once the people who will sit at the first table are chosen, the people who will sit at the second are automatically chosen. Therefore, the correct number of such arrangements is 
.
Since
,
then, setting 
,
,
the correct number of arrangements.
The number of ways to choose four people to sit at the first table without regard to the arrangement at either table is the number of combinations of four people from a set of eight; once the people who will sit at the first table are chosen, the people who will sit at the second are automatically chosen. Therefore, the correct number of such arrangements is
Since
then, setting
the correct number of arrangements.
A standardized test comprises five examinations: reading, writing, math, science, and social studies, and these examinations may be taken in any order; however, in one state, there is one restriction: neither the writing nor the math test may be taken last.
In how many different orders can the five tests be taken so that this restriction is met?
A standardized test comprises five examinations: reading, writing, math, science, and social studies, and these examinations may be taken in any order; however, in one state, there is one restriction: neither the writing nor the math test may be taken last.
In how many different orders can the five tests be taken so that this restriction is met?
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Because the last test cannot be either math or writing, the selection of an order in which the tests are taken must be looked at as a sequence of events as follows:
First, the test to be taken last must be one of three tests: reading, science, or social studies;
Second, the remaining four tests must be arranged in a particular order. There are four ways to schedule the first test, three remaining ways to schedule the third, two for the third, and one for the fourth.
By the multiplication principle, this makes
possible schedules.
Because the last test cannot be either math or writing, the selection of an order in which the tests are taken must be looked at as a sequence of events as follows:
First, the test to be taken last must be one of three tests: reading, science, or social studies;
Second, the remaining four tests must be arranged in a particular order. There are four ways to schedule the first test, three remaining ways to schedule the third, two for the third, and one for the fourth.
By the multiplication principle, this makes
possible schedules.
What is the probability that a person will roll an even number on a six-sided fair die?
What is the probability that a person will roll an even number on a six-sided fair die?
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In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:
The die has six sides with the following values: one tow, three, four, five, and six. Of these values the die has three that are even: two, four, and six. We can write the following probability.
Simplify.
Now, let's convert this into a percentage:
In order to solve this problem, we need to discuss probabilities. A probability is generally defined as the chances or likelihood of an event occurring. It is calculated by identifying two components: the event and the sample space. The event is defined as the favorable outcome or success that we wish to observe. On the other hand, the sample space is defined as the set of all possible outcomes for the event. Mathematically we calculate probabilities by dividing the event by the sample space:
The die has six sides with the following values: one tow, three, four, five, and six. Of these values the die has three that are even: two, four, and six. We can write the following probability.
Simplify.
Now, let's convert this into a percentage:
Two fair dice are tossed. You are not shown the dice but you are told that exactly one die shows an even number. Give the probability that a total of seven was thrown.
Two fair dice are tossed. You are not shown the dice but you are told that exactly one die shows an even number. Give the probability that a total of seven was thrown.
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Two fair dice can be thrown with 36 equally probable rolls. 18 rolls comprise one even number and one odd number. They are as follows - with the rolls that total seven in boldface:
6 of the 18 possible rolls have total 7, making the probability of a roll of 7, given that exactly one doe shows an even number, equal to 
.
Two fair dice can be thrown with 36 equally probable rolls. 18 rolls comprise one even number and one odd number. They are as follows - with the rolls that total seven in boldface:
6 of the 18 possible rolls have total 7, making the probability of a roll of 7, given that exactly one doe shows an even number, equal to 
Two fair dice are tossed. You are not shown the dice but you are told that at least one die shows an even number. Give the probability that a total of seven was thrown.
Two fair dice are tossed. You are not shown the dice but you are told that at least one die shows an even number. Give the probability that a total of seven was thrown.
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Two fair dice can be thrown with 36 equally probable rolls.9 rolls can be thrown that show two odd numbers: they are
This leaves 27 rolls with at least one even number, 6 of which result in a sum of 7:
This makes the probability of rolling a sum of 7
,
the correct choice.
Two fair dice can be thrown with 36 equally probable rolls.9 rolls can be thrown that show two odd numbers: they are
This leaves 27 rolls with at least one even number, 6 of which result in a sum of 7:
This makes the probability of rolling a sum of 7
the correct choice.
Find the mean of the following data set:
Find the mean of the following data set:
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The mean is found by adding the values in a set and dividing that number by the total number of values.
The mean is found by adding the values in a set and dividing that number by the total number of values.
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.
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.
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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.
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.
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.
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.
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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.
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 
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.