1. Week 6
Lecture 1
Chapter 10.
Sample Survey

2. Why Sample?

3. Why Sample? Examining the whole population may not be possible.
We hear this kind of thing all the time, a survey asking
“if there were provincial election tomorrow, which party would you vote for?”
How is this survey done?
Why is this survey done?
What happens if we tried to survey everybody?

4. Examine a Part of the Whole
The entire group of individuals that we want information about is called population.
We would like to know about an entire population of individuals, but examining all of them is usually impractical, if not impossible.
We settle for examining a smaller group of individuals – a sample – selected from the population.

5. Representative Sample
We would like the selected sample to be:
Random; members would have an equal chance of being selected.
For example if we want to understand university students’ experience at UTSC, we need to randomly select students from the entire population of UTSC students.
Sample need to be representative of the population:
Match the population
Avoid bias: over or underestimate

6. Sample Survey
Opinion polls are example of sample surveys, designed to ask questions of a small group of people in the hope of learning something about the entire population.
professional pollsters work quite hard to ensure that the sample they take is representative of the population.
If not, sample can be misleading information about the population.
Another example: I study students’ attitudes about statistics.
How: By administrating the Survey of Attitude Towards Statistics (SATS-36©) and linking students’ responses to their repository record from the Office of Registrar.

7. Why Does Randomization Work?
In short term, it is unpredictable. In long term, it is predictable.
We cannot predict which individuals are going to end up in sample.
With a large sample, the sample will have approximately right proportion of different genders, different age groups (e.g., young, old), different living areas (e.g., urban, rural), and of course many more layers (grouping) or things that we didn’t think of.

8. Three Keys of Sampling Examine a part of the whole (sample)
Randomize (to obtain the sample)
3. Sample Size

9. It’s the Sample Size
How large a random sample do we need for the sample to be reasonably representative of the population?
It’s the size of the sample, not the size of the population, that makes the difference in sampling.
Exception: If the population is small enough and the sample is more than 10% of the whole population, the population size can matter.
The fraction of the population that you’ve sampled doesn’t matter. It’s the sample size itself that’s important

10. Does a Census Make Sense?
A survey with all individuals in the population is called a census.
Wouldn’t it better to just include everyone and “sample” the entire population?
It can be difficult to complete a census:
sometimes the population changes due to changes in job locations for the individuals – thus, the findings might not be relevant to the current population by the time the census is completed.
there might be some individuals who are hard to locate or hard to measure, high cost, etc.
For example at the same time when the 2016 Canadian census were being conducted, there was a devastating fire in Fort McMurry in Alberta. Please see the below link regarding how the population was estimated (using municipal estimates) in Fort McMurry:

11. Populations and Parameters; Samples and Statistics
A parameter is a number that describes the population.
True values in the population.
Actual numerical values in the population.
E.g., Actual number of people voted for a political party in the entire country (entire population)
A parameter is a fixed number, but in practice we do not know its value.
We often use a statistic to estimate an unknown parameter.
A Statistic is a number that describes a sample.
The value of a statistic is known when we have taken a sample, but it can change from sample to sample.
E.g., a survey that is a subset of a population (NOT the entire population) may result in indicating the estimated number of people (in the survey) who voted for a candidate in a electoral political campaign.

12. Populations and Parameters; Samples and Statistics
The numerical values that we calculate from a sample, for example, the sample mean, sample standard deviation, and sample correlation, are statistics.
The statistics are estimates of population parameters (see table below):

13. Simple Random Sample
A simple random sample consists of n individuals from the population chosen in such a way that every set of n individuals has an equal chance to be the sample actually selected.
It requires a list of whole population (sampling frame).
Drawing a simple random sample:
Using random digit table
Using computer (a statistical software) – this is a sophisticated way.

14. Stratified Random Sampling
Divides population into separate groups, called strata.
Individual in each stratum are similar to each other (homogeneous).
Select a Simple Random Sample (SRS) from each stratum.
Combine the random selection from each stratum to make overall sample.
The strata are groups we want to compare.
Example:
 stratum 1
Stratum 2
Stratum 3

15. Example of Stratified Random Sampling
Suppose there are total of 960 students in grades 1, 2, and 3 in three schools.
We want to randomly select 100 students (combined).
The goal is to measure students’ reading abilities and compare them among the grade levels.
Here is what we know in terms of counts in each grade level (all three school combined):
Proportional Sampling:
Randomly select 40 students from grade 1 class;
Randomly select 30 students from grade 2 class;
Randomly select 30 students from grade 3 class.
Total sample size (n): = 100
Grade
Number of students
Percentage (of total) 1
400
400/960 * 100 ≅ 40% 2
280
280/960 * 100 ≅ 30% 3

16. Why Stratified Random Sampling is a Good Method?
Stratified random sampling can reduce bias.
Stratifying can also reduce the variability of our results.
Therefore, sample statistic should be closer to population parameter.

17. Cluster and Multistage Sampling
Cluster sampling is useful when:
a complete list of population is not available.
simple random sampling is difficult
we don’t have access to all strata or stratifying is not practical
Therefore, splitting the population into similar parts or clusters can make sampling more practical.
Divide the population into a number of clusters.
The goal is not to compare groups, but rather to use them to form a sample.
Randomly select number of clusters.
From each cluster, randomly select number of subjects (what and whom you want to sample).

18. Example of Cluster Sampling
Let’s say that I (researcher) would like to investigate high school students’ statistical literacy by the end of the Grade 12 Mathematics of Data Management Course.
This course is an introduction to statistics at the high school level in the Ontario Mathematics curriculum.
How can I do this research?
Where should I start?
Where is my randomly selected sample of students?

19. Example of Cluster Sampling
There are 42 public secondary schools in Mississauga. I will randomly select 3 schools. From each randomly selected school, I will randomly select 10 students who took Grade 12 Mathematics of Data Management Course; total sample size is 30 students.

20. Multistage Sampling Often hierarchy of clusters.
For example: chapter – section – sentence – word; we could choose:
Chapters
Sections within chosen chapters
Sentence within chosen section
Word within chosen sentence
The above example is an example of multistage sampling.
At each stage, the choice of selection is made by simple random sampling.

21. Example of Multistage Sampling
I could make my previous example a multistage sampling.
There are 51 cities in Ontario.
Randomly select cities from Ontario (e.g., Mississauga, Toronto).
From the randomly selected cities, randomly select board of educations (e.g., Peel District School Board, Dufferin-peel Catholic School Board, Toronto District School Board).
From the randomly selected boards of education, randomly select Secondary Schools.
From the randomly selected Secondary School, randomly select students who take the course:
Grade 12 Mathematics of Data Management.

22. In Summary .Choose cluster/multistage sampling for convenience.
Choose stratified random sampling for accuracy

23. Systematic Random Sampling
Sometimes we draw a sample by selecting individuals systematically.
For example, you might survey every 10th person on an alphabetical list of students.
To make it random, you must still start the systematics selection from a randomly selected individual.
When there is no reason to believe that the order of the list could be associated in any way with the responses sought, systematic random sampling can give a representative sample.

24. Example of Systematic Random Sampling
Suppose we want to systematically randomly select 100 students from 30,000.
N = 30,000 (population size); n = 100 (sample size).
We could use student directory (e.g., office of registrar) as our sampling frame.
System:
Let K be population size (N) divided by sample size (n).
K = N/n = 30,000/100 = 300
Select a subject at random from first K names in the sampling frame.
1. In this case, we randomly select one student from the first 300 students in the directory.
2. Select the second student from the next 300 students in the directory.
3. And so on, until we have 100 randomly selected students.

25. Things that can go wrong with Sample Survey
Not getting who you want (non-response)
E.g. Miss the mail in questionnaire
Solution: follow up with the potential cases (reminder messages)
Getting the question(s) right
Solution: Avoid favoring a certain answer in way the question is asked
Not giving choices for answer
E.g., getting open-ended responses
Solution: use Likert scale: strongly agree to strongly disagree
Sampling volunteers
Do not rely on people who choose to respond, e.g., callers to radio show
Sampling badly but conveniently
Undercoverage
Not being able to sample certain parts of population