CIE A Level Maths: Probability & Statistics 2复习笔记1.1.1 Sampling & Data Collection

What is a population, a sample and the difference between them?

• The population refers to the whole set of things which you are interested in
  • e.g. if a vet wanted to know how long a typical French bulldog slept for in a day then the population would be all the French bulldogs in the world
• A sample refers to a subset of the population which is used to collect data from
  • e.g. the vet might take a sample of French bulldogs from different cities and record how long they sleep in a day
• A sampling frame is a list of all members of the population
  • For example: a list of employees’ names within a company
• The sampling fraction is the size of the sample divided by the size of the population
• A population parameter is a numerical value which describes a characteristic of the population
  • These are usually unknown
  • For example: the mean height of all 16-year-olds in the UK
• A sample statistic is a value computed using data from the sample
  • These are used to estimate population parameters
  • For example: the mean height of 200 16-year-olds from randomly selected cities in the UK
• A census collects data about all the members of a population
  • For example: the Government in England does a national census every 10 years to collect data about every person living in England at the time

Why and how would I use a sample?

• A sample is used when it is impractical to use the population
  • e.g. It is impractical to test the lifetime of every lightbulb a factory produces as there would be none left to sell to customers!
• A sample can be used to estimate population parameters
  • e.g. Testing the lifetime of 50 lightbulbs a factory produces to estimate the average lifetime of all lightbulbs produced
• A sample needs to be representative of the population in order to produce reliable estimates of parameters – things to consider include
  • Sample size
    • A sample should be large enough
    • The sample size used will depend on the size of the population and the accuracy required

e.g.    A sample size of 30 to gauge the way the public will vote in a general election is clearly not large enough

• • Any bias in the data
    e.g. To estimate the proportion of people who enjoy watching movies, asking people in a cinema would introduce bias
  • Accidentally influencing data by phrasing of questions or the method of data collection

e.g.   “Do you agree that … ?” in a questionnaire is suggesting to the respondent that they should be agreeing
Collecting data about the length of vehicles by only collecting data about cars parked in a multi-storey car park

What is (simple) random sampling?

• Familiarity with random sampling and how to generate such a sample is expected
• For random sampling, every member of the population has an equal probability of being selected
  • A sampling frame is essential for a simple random sample
  • Simple random sampling is one method of random sampling - this is where every possible sample of size n has an equal probability of being selected as the sample
• In random sampling, every member of the population has a non-zero probability of being selected

How do I use random numbers or tables?

STEP 1  For simple random sampling, each member of the population – or each element in the sampling frame - is assigned a unique number

• • e.g.    The 50 members on the register of a local karate club are each allocated one of the numbers from 1 - 50

STEP 2   Random numbers – or more accurately, random digits - can be used to select elements from the sampling    frame

• • e.g.    Two-digit numbers would be needed for the karate club
  • Use pairs of valid digits from a list of random numbers, ignoring those that do not correspond to a member on the register
    If the string of random digits was 435231 then
    • 43 would be a valid member of the club
    • 52 would be ignored
    • 31 would be valid

• • Redundancy (the number of invalid random digits) can be reduced or eliminated by efficiently allocating which random numbers correspond to which member
    • e.g.   Assign the (100) numbers 00-99 to the 50 members of the karate club such that each member is represented by two possible numbers
    • So the first person on the register would be selected if 00 and 01 came up, the second person on the register would be selected if the 02 and 03 came up, and so on
  • Generally, calculators produce three digit (decimal) random numbers; tables are often printed with digits in blocks of five for ease of reading
    • as each digit is generated at random, they can be used to suit whatever the purpose is

Why not use a census instead of a sample?

• The main advantage of a census is that it gives fully accurate results
• The disadvantages of a census are:
  • It is time consuming and expensive to carry out
  • It can be impractical – for example it could destroy or use up all the members of a population when they are consumables (imagine a company testing every lightbulb they produce)
• Sampling creates a subset of the population from which data can be collected
• The advantages of sampling are:
  • It is quicker and cheaper than a census
  • It leads to less data needing to be analysed
• The disadvantages of sampling are:
  • It might not be representative of the population
  • It could introduce bias

What are the main criticisms of sampling techniques?

• The sampling technique will be described in a question
• Criticisms normally arise from the amount of data collected or the manner in which data is collected
  • Most sampling techniques can be improved by taking a larger sample so consider the sampling fraction
  • Sampling can introduce bias – so look to see if bias has been considered
    • A random sample helps to minimise bias
• A sample only gives information about the selected members
  • But the results can be used as estimates for the population
• Different samples may lead to different conclusions about the population