Describe what the following terms mean with regards to statistical analysis:
Sample
Population
For each of the following populations, give an example of a group of people that make up a sample of the population:
The population is all the students that attend the local high school.
The population is all people aged under 18, who live in Sydney.
The population is all people in a city who play in any organised sporting competition.
The population is all people who own a pet dog.
For each of the following samples, give an example of a population that the sample could have been chosen from:
A sample containing 50 people who drive white cars.
A sample of 50 people drawn from a population. In this sample, the youngest is 18 years old, and the oldest is 64.
A sample containing the first 50 people to enter a train station on a given day.
At a certain chocolate factory, 30\% of products contain nuts. 400 chocolates are tested to check if they meet the required quality for sale. Of those tested, 61\% contained nuts.
What is the population?
State the value of the population proportion.
State the value of the sample proportion.
Do the 400 chocolates tested represent a simple random sample?
State whether each of the following biased questions are leading or emotive:
Do you want a nutritious risotto for lunch or the usual sandwich?
Do you watch TV on a Sunday morning like everyone else?
I don’t like these shirts. Do you?
Do you prefer this rad shirt or the ordinary one on the shelves at the moment?
Explain why the following samples are biased:
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.
State whether the following questions are biased or fair:
Do you think the government should be allowed to cut down some of the oldest trees in the area to construct a metro railway line in the city?
Do you prefer newspapers or news on television?
Do you prefer the full time degree program or part time degree program?
Should the government enforce a minimum drinking age for its citizens?
Do you eat at least the recommended number of servings of fruits and vegetables to ensure a healthy and long life?
Do you think bike helmets should be mandatory for all bike riders?
Do you prefer the natural beauty of hardwood floors in your home?
Do you exercise regularly?
Do you feel that the TV news is an inaccurate portrayal of life’s problems?
Don't you think this newspaper is biased?
Do you prefer the look and feel of thick lush carpeting in your living room?
Do you take these extra strength multi-vitamins to supplement your diet?
State whether the following scenarios use biased sampling methods:
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.
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.
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 school principal wants to estimate the number of students who ride a bicycle to school. State whether the following samples would avoid bias:
All students who are in the school band.
Eight students in the hallway.
Ten students from each grade, chosen at random.
130 students during the lunch periods.
The Skin Cancer Council wants to survey the population to approximate the average amount of time someone spends in the sun each day.
Determine whether the following methods could minimise completion bias in the survey responses:
Requiring responders to note the exact times of the day that they spend in the sun.
Calling people during standard work hours.
Making sure the survey questions are comprehensive by having many questions.
Having one short question where responders select from given ranges of values for the number of hours they spend in the sun.
To determine which political party is most likely to win in an upcoming election, a sample of 500 people is to be chosen and asked who they will vote for.
Determine whether the following sampling techniques result in selection bias:
Selecting 500 people randomly at a local shopping centre.
Selecting 500 people randomly from the national census.
Selecting 500 parents randomly after they pick up their children from school.
Selecting the first 500 people who walk into an office building.
Some students want to conduct an interview to find out the amount of time students spend doing homework each week. They brainstorm methods on how to collect a random sample of students for the interview.
Determine whether the following methods would involve selection bias:
Ask the first 80 students who walk in to the Library.
Wait at the entrance of the school and ask the first 100 students who arrive before 7 am to avoid disruption to the school day.
At school assembly, randomly select 70 students to be interviewed.
Determine whether the following methods would involve self-selection bias:
Leave a 'nomination sheet' in the library and ask only those people who write their names on it.
Ask everyone in Year 8.
Call a meeting of all students who are interested in taking part and ask all the people who attend the meeting.
Describe how you would conduct the survey to avoid selection bias and self-selection bias.
A research organisation wants to determine tourists' impressions of Australia. They create a survey consisting of several questions. Which of the following questions encourages bias? Explain your choice.
Do you think it is too expensive to travel around Australia?
How much time did you spend in Australia?
What was your most memorable experience in Australia?
What cities did you visit in Australia?
A study is to be conducted to research how sugar affects brain activity. Determine whether the following could result in biased conclusions:
Conducting the study in a country known for high rates of refined food intake.
Having the study funded by a major soft drink manufacturer as they would be able to donate significant funds to the research.
Having the study conducted by a government health agency funded only by the federal government.
Conducting a double blind study where half the participants are in a control group, and the other half are given a treatment. The researchers and participants don't know who is in which group until after the experiment.
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 groups of 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.
Only people who are employed.
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.
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 50 of the 100 fishes known to exist in the area.
Did they catch a large enough proportion to make a reliable conclusion?
What else could be done to make sure the sample is representative of the population?
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.
Explain why this is not an appropriate way to conduct a poll.
Laura is a newsagent. Her shop is next to a train station. She wants to find out how many people who use the train station enter her shop every week. She decides that at 9 am on a Sunday morning, she will count how many of the first 10 people she sees in the station walk into her shop.
Explain why this would produce a biased result.
A political polling company calls 1000 people at home between 4 pm and 7 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 410 people who answered their call.
How could they have used random sampling to choose the 1000 people to call?
Explain why the company's published results will not be accurate.
Consider a fair 8-sided die with faces labelled from 1 to 8. Let X be the outcome when the die is rolled.
Complete the table of values for the probability distribution for X:
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| P(X=x) |
Calculate the mean of the distribution.
Calculate the standard deviation of the distribution correct to two decimal places.
The die was rolled 20 times with the following results:8, \, 3, \, 3, \, 5, \, 4, \, 8, \, 1, \, 7, \, 6, \, 5, \, 2, \, 2, \, 3, \, 5, \, 4, \, 3, \, 8, \, 6, \, 5, \, 1
Calculate the sample mean of the results.
Calculate the sample standard deviation to two decimal places.
The random variable X is uniformly distributed with a variance of \dfrac{25}{12} over the interval \\ 1 \leq x \leq 6.
Calculate the mean for X.
Calculate the standard deviation for X to three decimal places.
A sample of size 10 was taken from the distribution:3.92, \, 4.47, \, 1.73, \, 3.87, \, 1.58, \, 2.54, \, 1.53, \, 5.99, \, 2.48, \, 3.06
Calculate the mean of the sample.
Calculate the standard deviation of the sample to three decimal places.
R is a binomial variable with n = 14 and p = 0.25. Two samples, A and B, each of size 10, are taken from R are shown below:
A: \, 4, \, 4, \, 2, \, 4, \, 2, \, 2, \, 5, \, 5, \, 4, \, 5
B: \, 4, \, 2, \, 3, \, 2, \, 1, \, 7, \, 4, \, 2, \, 4, \, 5
Calculate the theoretical mean of R.
Calculate the theoretical standard deviation of R to four decimal places.
Calculate the mean for sample A.
Calculate the mean for sample B.
Calculate the standard deviation for sample A to three decimal places.
Calculate the standard deviation for sample B to three decimal places.
Which sample is more like the population?
The normal variable X has a mean of 120 and a standard deviation of 15. Two samples, A and B, each of size 10, are taken from X are shown below:
A: \, 141.88, \, 131.53, \, 126.36, \, 108.49, \, 116.79, \, 123.34, \, 110.09, \, 90.37, \, 115.13, \, 123.46
B: \, 121.04, \, 116.66, \, 108.68, \, 130.62, \, 106.74, \, 134.58, \, 108.83, \, 111.65, \, 131.4, \, 133.1
Calculate the mean for sample A.
Calculate the mean for sample B.
Calculate the standard deviation for sample A to three decimal places.
Calculate the standard deviation for sample B to three decimal places.
Which sample is more like the population?
X is a discrete uniform distribution across the integers 1, 2, 3, 4, 5, 6 and 7.
Calculate the mean and standard deviation of the distribution.
A sample of size 25 is taken from this distribution and the graph and table of results are shown below:
| Value | Frequency |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 7 |
| 4 | 3 |
| 5 | 4 |
| 6 | 3 |
| 7 | 2 |
Calculate the mean and standard deviation of this sample to two decimal places.
A sample of size 100 is taken from this distribution and the graph and table of results are shown below:
| Value | Frequency |
|---|---|
| 1 | 16 |
| 2 | 7 |
| 3 | 13 |
| 4 | 21 |
| 5 | 18 |
| 6 | 13 |
| 7 | 12 |
Calculate the mean and standard deviation of this sample to two decimal places.
Another sample of size 100 is taken from this distribution and the graph and table of results are shown below:
| Value | Frequency |
|---|---|
| 1 | 19 |
| 2 | 16 |
| 3 | 8 |
| 4 | 21 |
| 5 | 18 |
| 6 | 13 |
| 7 | 12 |
Calculate the mean and standard deviation of this sample to two decimal places.
X is a Bernoulli distribution with P \left( X=0 \right) = 0.21 and P \left( X=1 \right) = 0.79.
Calculate the mean and standard deviation of the distribution to two decimal places.
A sample of size 25 is taken from this distribution and the graph and table of results are shown below:
| Value | Frequency |
|---|---|
| 0 | 6 |
| 1 | 19 |
Calculate the mean and standard deviation of this sample to two decimal places.
A sample of size 100 is taken from this distribution and the graph and table of results are shown below:
| Value | Frequency |
|---|---|
| 0 | 18 |
| 1 | 82 |
Calculate the mean and standard deviation of this sample to two decimal places.
A sample of size 100 is taken from this distribution and the graph and table of results are shown below:
| Value | Frequency |
|---|---|
| 0 | 22 |
| 1 | 78 |
Calculate the mean and standard deviation of this sample to two decimal places.
X is a normal distribution with mean 50 and standard deviation 8.
A sample of size 25 is taken from this distribution and the graph and table of results are shown below:
| Value | Frequency |
|---|---|
| 26 | 0 |
| 34 | 0 |
| 42 | 6 |
| 50 | 11 |
| 58 | 5 |
| 66 | 3 |
| 74 | 0 |
Calculate the mean and standard deviation of this sample to two decimal places.
A sample of size 100 is taken from this distribution and the graph and table of results are shown below:
| Value | Frequency |
|---|---|
| 26 | 0 |
| 34 | 8 |
| 42 | 30 |
| 50 | 32 |
| 58 | 22 |
| 66 | 7 |
| 74 | 1 |
Calculate the mean and standard deviation of this sample to two decimal places.
A sample of size 100 is taken from this distribution and the graph and table of results are shown below:
| Value | Frequency |
|---|---|
| 26 | 0 |
| 34 | 8 |
| 42 | 23 |
| 50 | 31 |
| 58 | 26 |
| 66 | 12 |
| 74 | 0 |
Calculate the mean and standard deviation of this sample to two decimal places.
By considering the results of the previous three questions, describe what happens as we take a larger and larger sample of a population with regards to the graph of the data, the mean and the standard deviation.
Consider a spinner with ten equal segments, numbered 1 to 10. Let X be the number the spinner lands on.
Complete a probability distribution table for X.
Calculate the mean of the distribution.
Calculate the standard deviation of the distribution correct to two decimal places.
Simulate 50 spins of the spinner using your CAS calculator and calculate the mean and standard deviation of your sample.
Simulate 100 spins of the spinner using your CAS calculator and calculate the mean and standard deviation of your sample.
Compare and contrast your two samples with regards to the shape and properties of X.
X is uniformly distributed over the domain 12 to 20.
Calculate the mean of the distribution.
Calculate the standard deviation of the distribution correct to two decimal places.
Simulate sampling 50 times from the distribution using your CAS calculator and calculate the mean and standard deviation of your sample.
Simulate sampling 120 times from the distribution using your CAS calculator and calculate the mean and standard deviation of your sample.
Compare and contrast your two samples with regards to the shape and properties of X.
X is uniformly distributed over the domain 20 to 30.
Calculate the mean of the distribution.
Calculate the standard deviation of the distribution correct to two decimal places.
Simulate sampling 50 times from the distribution using your CAS calculator and calculate the mean and standard deviation of your sample.
Simulate sampling 120 times from the distribution using your CAS calculator and calculate the mean and standard deviation of your sample.
Compare and contrast your two samples with regards to the shape and properties of X.
X is normally distributed with a mean of 80 and a standard deviation of 7.
Simulate sampling 50 times from the distribution using your CAS calculator and calculate the mean and standard deviation of your sample.
Simulate sampling 120 times from the distribution using your CAS calculator and calculate the mean and standard deviation of your sample.
Compare and contrast your two samples with regards to the shape and properties of X.
X is normally distributed with a mean of 150 and a standard deviation of 15.
Simulate sampling 100 times from the distribution using your CAS calculator and sketch the resulting histogram of your sample.
Simulate sampling 150 times from the distribution using your CAS calculator and sketch the resulting histogram of your sample.
Compare and contrast the graphs of your two samples with regards to the graph of X.
X is a binomial random variable with n=25 and p=0.3.
Calculate the mean for X.
Calculate the standard deviation for X correct to two decimal places.
Simulate sampling 50 times from the distribution using your CAS calculator and calculate the mean and standard deviation of your sample.
Simulate sampling 120 times from the distribution using your CAS calculator and calculate the mean and standard deviation of your sample.
Compare and contrast your two samples with regards to the shape and properties of X.
X is a binomial random variable with n=15 and p=0.75.
Calculate the mean for X.
Calculate the standard deviation for X correct to two decimal places.
Simulate sampling 100 times from the distribution using your CAS calculator and sketch the resulting histogram of your sample.
Simulate sampling 120 times from the distribution using your CAS calculator and sketch the resulting histogram of your sample.
Compare and contrast the graphs of your two samples with regards to the graph of X.