11.01 Collecting and displaying data
After the government decided to increase the minimum retirement age, a news poll selected a group of people to ask their opinions on the changes.
Determine whether the following people should be represented to avoid sampling bias.
People in the community who have a wide variety of views, excluding politicians and policy makers.
People in the community who have a wide variety of views, even if they are not directly affected by the changes.
Only people in the community who would be directly affected by the changes.
A radio station conducts a poll asking its listeners to call in to say if they are for or against restrictions on scalpers selling tickets for gigs at a higher price.
Determine whether the following are the reasons why this is not an appropriate way to conduct a poll.
A large variety of people are likely to call.
A person can call more than once, so they could be counted more than once.
People with stronger views are more likely to call than those who don’t have a strong view.
A question is inappropriate for a survey given it fulfills any of the following criteria:
Question is unclear
Emotional language
Should be a "yes" or "no" question
Question is too personal
Using the criteria list, determine why each of the questions are inappropriate.
How many large electronic devices are in your home?
Many people have worked incredibly hard and even died making this bridge. Do you like the bridge?
What is your favourite breed of dog?
Determine whether the following questions are likely to give biased results or not:
Yvonne is asking people on her soccer team, "What's your favourite sport?"
Lachlan randomly selected people from his school to find about the school sports. He asked "What's your favourite school sport?"
Tricia randomly selected people from her school and asked, "The local AFL team is donating money to our school this term. What's your favourite sport?"
Lulu is interviewing people outside of a train station. She asks "Should the government spend more money on public transportation?"
Shadie is interviewing people about how shark nets are affecting shark attack numbers. He asks "Shark nets cost \$1\,000\,000 per year and barely help. Do you support shark nets?"
Write a possible question you could ask in order to find out the following information:
You want to gather information about how many people have had a dog as a pet.
You're interviewing students to find out how often they read.
For each of the following scenarios, construct a frequency table for the results:
Mr. Rodriguez recorded the number of pets owned by each of the students in his class. He found that 15 people had no pets, 19 people had one pet, 3 people had two pets and 8 people had three pets.
Noah asked his students to choose their favourite method of travelling. 9 students picked car, 13 people picked plane, and 5 people picked boat.
Rosey asked 15 of her friends what their favourite colour is. Their responses were as follows:
\text{pink},\, \text{blue},\, \text{yellow},\, \text{blue},\, \text{green},\, \text{pink},\, \text{blue},\, \text{blue},\, \\ \text{blue},\, \text{pink},\, \text{pink},\, \text{blue},\, \text{green},\, \text{pink},\, \text{blue}
Mrs. Brown asked her students how many bedrooms they have in their house. Their responses were as follows:
4,\, 4,\, 4,\, 3,\, 4,\, 3,\, 3,\, 5,\, 5,\, 4,\, 4,\, 3,\, 3,\, 4,\, 4,\, 4,\, 3,\, 4,\, 5,\, 4
Mr. Smith asked his students how many siblings they have. Their responses were as follows:
1,\, 1,\, 1,\, 2,\, 2,\, 1,\, 0,\, 3,\, 0,\, 1,\, 2,\, 1,\, 1,\, 2,\, 2,\, 3,\, 1,\, 2,\, 3,\, 3,\, 1,\, 2,\, 2,\, 1,\, 2
A gymnast received the following scores over several rounds of back to back competition:
Given that she received these scores over 15 rounds of competition, find the value of A.
State the percentage of rounds for which the gymnast received a score of 8. Round your answer to two decimal places.
State the percentage of rounds the gymnast received a score less than 7. Round your answer to two decimal places.
| Score | Frequency |
|---|---|
| 5 | 4 |
| 6 | 1 |
| 7 | 4 |
| 8 | A |
| 9 | 2 |
A survey was conducted which asked 30 people how many books they had read in the past month. Based on the frequency table provided, state whether the following statements are correct:
11 people have read between 6 and 10 books in the past month.
28 people have read at most 15 books in the past month.
We cannot determine from the table how many people have read exactly 12 books.
We can determine that 2 people have read exactly 5 books in the past month.
| Number of books read | Frequency |
|---|---|
| 1-5 | 2 |
| 6-10 | 11 |
| 11-15 | 15 |
| 16-20 | 2 |
The set of marks for a class of students is given below:
81,\quad 81,\quad 71,\quad 81,\quad 81,\quad 61,\quad 71,\quad 93,\quad 71,\quad 58,\quad 71,\\ 58,\quad 61,\quad 93,\quad 93,\quad 71,\quad 61,\quad 61,\quad 81,\quad 58,\quad 93
Organise the data into a frequency table.
Find the total number of students in the class.
Find the number of students who will get a Distinction grade \left( 80 \lt \text{Score} \leq 90 \right).
Find the number of students who will get a High Distinction grade \left( \text{Score} \gt 90 \right).
Find the percentage of students obtaining a High Distinction grade. Round your answer to two decimal places.
Consider the following set of scores:
12,\quad 59,\quad 61,\quad 27,\quad 58,\quad 18,\quad 76,\quad 27,\quad 52,\quad 19,\quad 13,\quad 56,\quad 71,\quad 31,\quad 73, \\ 60,\quad 41,\quad 17,\quad 22,\quad 68,\quad 57,\quad 15,\quad 40,\quad 19,\quad 76,\quad 44,\quad 60,\quad 55,\quad 36
Complete the grouped frequency table for the data.
State the modal class for the data.
| \text{Score } (x) | \text{Frequency } (f) |
|---|---|
| 10 \to 19 | |
| 20 \to 29 | |
| 30 \to 39 | |
| 40 \to 49 | |
| 50 \to 59 | |
| 60 \to 69 | |
| 70 \to 79 |
This stem-and-leaf plot records the ages of customers at a beachside cafe last Sunday.
Complete the frequency table for this data:
| Age | Frequency |
|---|---|
| 10-19 | |
| 20-29 | |
| 30-39 | |
| 40-49 | |
| 50-59 | |
| 60-69 |
| Leaf | |
|---|---|
| 1 | 0\ 4\ 7 |
| 2 | 1\ 4\ 5\ 7 |
| 3 | 1\ 3\ 9 |
| 4 | 1\ 3\ 5\ 6\ 8\ 9 |
| 5 | 4\ 5\ 6\ 7\ 8\ 9 |
| 6 | 0\ 2\ 3\ 6 |
Key: 5 | 2 \ = \ 52
A city council selected a number of houses at random. They determined the fastest travel time (in minutes) from each house to the nearest hospital, and recorded the following results:25, \, 37, \, 16, \, 27, \, 27, \, 35, \, 21, \, 18, \, 19, \, 49, \, 14, \, 19, \, 31, \, 42, \, 18
Represent this data as an ordered stem-and-leaf plot.
The back-to-back stem-and-leaf plot shows the batting scores of two cricket teams, the Grasshoppers and the Cicadas:
What is the highest score for the Grasshoppers?
What is the highest score for the Cicadas?
| Grasshoppers | Cicadas | |
|---|---|---|
| 5\ 2 | 3 | 1\ 5\ 6\ 6\ 9 |
| 9\ 7\ 6\ 4\ 4\ 1 | 4 | 1\ 5 |
| 8\ 1 | 5 | 5\ 6 |
| 6 | 3 |
Key: 6|1|2=12 \text{ and } 16
The back-to-back stem plot shows the number of pieces of paper used over several days by Paul’s and Edward’s students:
Determine whether the following statements is true.
Paul's students did not use 7 pieces of paper on any day.
Edward's median is higher than Paul’s median.
The median is greater than the mean in both groups.
| Paul | Edward | |
|---|---|---|
| 7 | 0 | 7 |
| 3 | 1 | 1\ 2\ 3 |
| 8 | 2 | 8 |
| 4\ 3 | 3 | 2\ 3\ 4 |
| 7\ 6\ 5 | 4 | 9 |
| 3\ 2 | 5 | 2 |
\text{Key: 6|1|2 \ = \ 16 \text{ and } 12}
The data shows the results of a survey conducted on the price of concert tickets locally and the price of the same concerts at an international venue:
What was the most expensive ticket price at the international venue?
What was the median ticket price at the international venue?
What percentage of local ticket prices were cheaper than the international median?
At the international venue, what percentage of tickets cost between \$90 and \$110 (inclusive)?
At the local venue, what percentage of tickets cost between \$90 and \$100 (inclusive)?
| Local | International | |
|---|---|---|
| 7\ 6\ 2\ 2 | 6 | 3\ 7 |
| 9\ 7\ 6\ 3\ 0 | 7 | 3\ 4\ 6\ 8 |
| 8\ 5\ 5\ 3\ 0 | 8 | 0\ 2\ 6\ 9 |
| 7\ 7\ 4\ 1\ 0 | 9 | 1\ 1\ 6\ 6\ 7 |
| 5 | 10 | 1\ 3\ 3\ 7\ 9 |
Key: 6|1|2 = \$16 \text{ and } \$12
The stem-and-leaf plot shows the test scores of a school's two Year 7 classes, A and B
What is the highest score in Class A?
What is the highest score in Class B?
Find the mean score of class A, correct to two decimal places.
Find the mean score of class B, correct to two decimal places.
Calculate the overall mean of the year 7 students, correct to two decimal places.
| Class A | Class B | |
|---|---|---|
| 3\ 2 | 6 | 3 |
| 0\ 6\ 6 | 7 | 2\ 5\ 5 |
| 7\ 3 | 8 | 1\ 4\ 8 |
| 9 | 5 |
\text{Key: 6|1|2 = 12 \text{ and } 16}
This column graph shows the star ratings by different patrons at a local hotel:
Which rating was the most common?
How many ratings of 5 were there?
How many more people gave a rating of 3 than a rating of 2?
State the modal rating.
Laura has 7 red marbles, 3 green marbles, 9 yellow marbles, 14 black marbles, and 7 blue marbles.
Construct a column graph to represent this information.
A director wants to see what a small group of people think of their film. The audience is asked to give the film a score between 0 and 5. The results are shown in the histogram below:
Given that the median and mean are both 2.5 and the mode is 0, determine whether the director can make the following conclusions:
The mean is 2.5 so most people thought the movie was ok.
People were very divided by the movie.
The median was 2.5 so most people thought the movie was ok.
The mode is 0 so everyone disliked this film.
Consider the following histogram:
Determine the mode.
How many scores of 3 were there?
How many more scores of 1 were there than scores of 3?
Data is represented in a histogram as shown:
Complete the following frequency table:
| Score | Frequency |
|---|---|
| 20 | |
| 22 | |
| 24 | |
| 26 | |
| 28 | |
| 30 |
State the mode(s) of the data.
A government agency records how long people wait on hold to speak to their representatives. The results are displayed in the following histogram:
Complete the following frequency table:
| Length of hold (minutes) | Frequency |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 |
How many phone calls were made?
How long in total did these people wait on hold?
Find the mode of the wait times.
Construct a histogram to represent the following data:
43, 47, 42, 45, 46, 45, 43, 44, 43, 42, 46, 43, 44, 47, 46, 44, 44, 43, 47, 42, 47, 43, 43, 47
The amount of snowfall (in centimetres) is recorded at the base of the mountain each day.
To create a frequency histogram of the data, which values go on the horizontal axis: the amount of snowfall or the number of days it snowed each amount?
The snowfall recorded each day, to the nearest centimetre, is as follows:
0, 3, 5, 2, 0, 2, 3, 2, 1, 4, 3, 3, 4, 4, 3, 6, 1, 2, 6, 2
Construct a frequency histogram of the data.
On how many days did 6 centimetres of snow fall?
On how many days did at least 3 centimetres of snow fall?
State the mode(s) for the amount of snowfall.
The histogram shows the average load times for fifty two webpages:
Use the data from the histogram to complete the following frequency table:
| Load time (seconds) | Frequency |
|---|---|
| 0-3 | |
| 3-6 | 13 |
| 6-9 | |
| 9-12 | 7 |
| 12-15 | 4 |
| 15-18 | |
| 18-21 | |
| \text{Total} | 52 |
The ages of new mothers in a particular hospital are displayed on the following histogram:
| Age | Frequency |
|---|---|
| 28 \leq \text{Age} \ <30 | |
| 30 \leq \text{Age} \ <32 | |
| 32 \leq \text{Age} \ <34 | |
| 34 \leq \text{Age} \ <36 | |
| 36 \leq \text{Age} \ <38 | |
| 38 \leq \text{Age} \ <40 |
Complete the frequency table based on the histogram.
How many new mothers were there?
How many mothers were under 34 years old?
How many mothers were 36 years old or older?
The frequency table shows the average time spent travelling to work for fifty two people:
Construct a histogram to display the data shown in the frequency table.
| Commute time (min) | Frequency |
|---|---|
| 0 \leq x \lt 20 | 15 |
| 20 \leq x \lt 40 | 17 |
| 40 \leq x \lt 60 | 10 |
| 60 \leq x \lt 80 | 6 |
| 80 \leq x \lt 100 | 4 |
| \text{Total} | 52 |
Determine whether each of the following statements about the data is accurate:
The data suggests that people don't care too much about how far away from work they live. Roughly equal portions of people live less than 40 minutes away and more than 40 minutes away.
The data suggests that people prefer a shorter commute to work. A majority live within 40 minutes travel, and in general the longer the commute the less people there are in that category.
The data shows that everyone lives within an hours travel from their work, with the peak amount of people living between 20 and 40 minutes away.
The data shows that most people travel to work by car or by walking, since most travel times are fairly short, and only a few people travel by bus or train.
The frequency table shows the price of the most recent book that sixty four university students bought:
Construct a histogram to display the data shown in the frequency table.
| \text{Price }(\$) | \text{Frequency} |
|---|---|
| 0 \leq x \lt 20 | 3 |
| 20 \leq x \lt 40 | 18 |
| 40 \leq x \lt 60 | 9 |
| 60 \leq x \lt 80 | 5 |
| 80 \leq x \lt 100 | 20 |
| 100 \leq x \lt 120 | 9 |
| \text{Total} | 64 |
Determine whether each of the following statements about the data is accurate:
The data shows that every university student has bought a book in the last couple of weeks, with some even spending over \$100 on a book.
The data shows that the average amount spent on a book by a university student is between \$20 and \$39, with not many students spending more than \$60 on a book.
The data shows that the average amount spent on a book by a university student is between \$80 and \$99. This suggests that most students purchased an expensive textbook recently.
The data shows that there are two main price ranges of books that were recently bought by university students. One peak is between \$20 and \$39, the other between \$80 and \$99.