11.04 Samples from a population
State whether each of the following is an instance of a sample or a census:
A random selection of some people at a mall.
A stock take of all the goods in store.
A crash test of new cars just manufactured by a factory.
Asking all the teachers at your school whether they approve of a new class timetable.
An election to decide the premier of Queensland.
Asking a random selection of students in your class whether they approve of the teacher.
A taste test of a large batch of cookies you have just baked.
A body scan of randomly selected passengers at Melbourne International Airport.
Determine whether the following is a census or sample.
Lucy has asked everyone in her office what snacks should be provided in the office.
James asks a few of his friends how they did in the test to see if he is above average in his class.
Joanne finds the height of the entire class to try to find the average height of 15 year old students in Australia.
Justin has determined the age of 10\% of houses from each suburb in Sydney.
Valentina tests every engine that the factory produces.
Oprah checks every dog brought to her vet to assess the treatment of dogs in the city.
Determine whether a census or sample would be better for the following examples.
A school teacher wants to find out what grade level his students are reading at.
A university student wants to find the average reaction times for humans.
A hospital wants to find out which heart rate machines are faulty.
A store manager wants to know the exact number of stolen items from the store.
Tricia wants to make sure the fireworks that she creates explode properly.
A farmer wants to find the average height of the stalks of wheat in his field.
State whether a sample or a census would be more suitable for determining:
The number of ambulances in Australia.
The number of people watching Lost.
The average number of letters in the surnames of teachers at your school.
The number of smokers in Queensland.
The average weight of students in your class.
The average dress size of girls in Germany.
The number of students with brown eyes in a library.
The most listened to radio station in Sydney.
The average handwriting speed of students in Victoria.
The average time it takes students in your grade to run a lap around the school.
The number of train stations in Sydney.
The amount of petrol used up by the family car in December.
Students' favourite colour in your class.
The average time it takes students in your class to sprint 100 metres.
The number of post offices in Western Australia.
The number of dogs in Queensland.
For each item, detemine the type of sampling method for these following events
Drawing out the winning ticket number in a lottery .
Choosing every 5th person on the class roll to take part in a survey .
Choosing 10\% of the students in each year for Years 7-12.
At a music concert there is expected to be a crowd of 3240 people. The band is giving away some shirts for some people that attend based on what order they enter the venue.
If the band are to give away 18 shirts, find the proportion in simplified fraction of people that will get shirts.
If the band are to give away 60 shirts, find the proportion in simplified fraction of people that will get shirts.
A factory produces 1820 TVs every day. How many TVs are tested daily if the factory tests a systematic sample of:
every 13th TV
every 70th TV
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, even if they are not directly affected by the changes.
People in the community who have a wide variety of views, excluding politicians and policy makers.
Only people in the community who would be directly affected by the changes.
Only people who are employed.
The owner of a movie cinema wants to use stratified sampling in their survey to people who watch a movie. Determine whether they can use the following systematic sampling methods:
Interview 10\% of the people from both the people that used the candy bar and 10\% of people who didn't.
Interview 10\% of the people from each movie.
Interview every 9th person that purchases a ticket.
Interview every person that sees a horror movie.
The principal wants to use systematic sampling in their survey on how students in their school feel about the new sports equipment. Determine whether the following are systematic sampling methods:
Interview 150 random people from the year.
Organise every student in the year by their height and then select every 7th person in the line.
Interview every 7th person on the school student list for that year grade.
Interview every 7th person that walks past their office.
Interview one random person from each class in that year.
Irene is interested in which students from her school catch public transport. Determine whether the following sampling methods are likely to be biased or not. Explain your answer.
Selecting every 10th person on the bus she catches.
Selecting every 10th person on the student list.
Selecting the first 50 students that arrive in the morning.
Selecting by having a computer randomly choose student numbers.
For each item, consider the following events to answer:
Is the sample chosen biased or fair?
If biased, identify the type of bias involved.
Hannah is surveying customers at a shopping precinct. She wants to know which stores customers shop at the most. She walks around an entertainment store and chooses 30 customers from the store for the survey.
A TV station wants to know what the most popular type of music is, so they ask listeners to contact them and vote for their favourite type of music.
The community health nurse wants to survey the students in a school about their eating habits. At lunchtime, she stands by a vending machine and surveys every student who purchases something from the machine.
Sean wants to know what 8th graders think of their English class, so he polls 70 random 8th graders.
Is the sample chosen biased or fair?
A school principal wants to estimate the number of students who ride a bicycle to school. Which sample(s) should be used to not introduce bias?
S1: All students who are in the school band.
S2: Eight students in the hallway.
S3: Ten students from each grade, chosen at random.
S4: One hundred thirty students during the lunch periods.
Which of these scenarios uses biased sampling methods?
Scenario 1: A community nurse wants to know the average height of all 7th graders that attend the school where she visits, so she measures the height of all the basketball players.
Scenario 2: A city councilman asks members of the ice hockey team if they would prefer a new skateboard park or a new ice-skating rink to be built as the new building project.
Scenario 3: The lifeguard of a water park wants to determine which water rides are enjoyed the most so he asks every tenth person who leaves the park to list their three favourite rides.
A political polling company calls 1000 people at home between 4 pm and 6 pm on weeknights to find out who they are most likely to vote for in an upcoming election. They publish their numbers based on the responses of only the 450 people who answered their call.
How could they use random sampling to choose the 1000 people to call?
Explain the fault with the method the company used.
Marine biologists want to determine if a local species of fish is growing to a smaller size than it used to. They collect and measure 150 of the 450 fish known to exist in the area.
Did they catch a large enough proportion of the species to be able to determine if the fish are growing to a smaller size?
What additional information would make us more confident that the sample is representative of the population?
The local mayor wants to determine how people in her town feel about the new construction project. Determine the type of sampling each scenario uses:
Selecting every 50th name from an alphabetical list of residents.
Giving each resident a random number between 1 and 10 and then selecting everyone with the number 3.
Selecting 10\% of the residents from each suburb.
At a certain university, 12\% of students study engineering. 2000 random students have been asked what subject they are studying. Of those asked, 14\% were engineering students.
What is the population?
Find the value of:
The population proportion
The sample proportion
Do the 2000 students tested represent a simple random sample?
At a certain chocolate factory, 40\% of products contain nuts. 200 chocolates are tested to check if they meet the required quality for sale. Of those tested, 64\% contained nuts.
What is the population?
Find the value of:
The population proportion
The sample proportion
Do the 200 chocolates tested represent a simple random sample?
A group of people is divided into four teams - Blue, Red, Green and Yellow. The table shows the number of people in each team:
How many people are there combined in all of the teams?
If a stratified sample of 1 in 30 is to be taken from the group, state the number of people who will be chosen.
For the sample to be stratified, give the number of people should be chosen from the following:
Blue team
Red team
Green team
Yellow team
| Team | Number of People |
|---|---|
| Blue | 150 |
| Red | 390 |
| Green | 270 |
| Yellow | 300 |
In a group of 1600 students, 600 are male and 1000 are female. A stratified sample of 16 is to be selected from the group based on gender.
How many males should be selected?
How many females should be selected?
A factory produces 3564 DVD players every day. How many DVD players are tested daily if the factory tests a systematic sample of:
Every 18th DVD player.
Every 66th DVD player.
Out of 1638 students in a school, 234 were chosen at random and asked their favourite primary colour. From those students, 93 choose red, 42 blue and 99 yellow.
One in every how many students at the school was sampled?
Estimate the total number of students in the whole school who prefer the colour:
Red
Blue
Yellow
In a group of 480 students, 120 are primary students and 360 are secondary students. A stratified sample of 120 is to be selected from the group based on year level.
How many primary students should be selected?
How many secondary students should be selected?
Users of a particular streaming service can be in one of four categories - Standard, Family, Premium or Business. The table shows the number of people in each category:
How many customers are there across all the categories?
If a stratified sample of 600 is to be taken from the group, what proportion of people will be chosen?
For the sample to be stratified, find the number of customers that should be taken from the each category:
Standard
Family
Premium
Business
| Team | Number of People |
|---|---|
| \text{Standard} | 3500 |
| \text{Family} | 1500 |
| \text{Premium} | 3000 |
| \text{Business} | 2000 |
The length of King George Whiting fish found off the coast of South Australia are known to be normally distributed with a mean of 29 \text{ cm} and a standard deviation of 2 \text{ cm}. Samples of four various fish species are submitted to the fisheries department, but the name of each species has been lost.
Which sample is most likely to represent those of the King George Whiting?
Fish 1: \, 37.29, \, 33.15, \, 32.96, \, 34.89, \, 35.11, \, 33.12, \, 32.84
Fish 2: \, 26.32, \, 23.61, \, 25.42, \, 25.07, \, 25.23, \, 25.59, \, 23.49
Fish 3: \, 31.12, \, 28.16, \, 25.36, \, 30.05, \, 26.2, \, 28.99, \, 29.14
Fish 4: \, 41.54, \, 28.11, \, 28.76, \, 27.62, \, 18.29, \, 33.41, \, 31.96