Mathematics in the Modern World

• Learning Outcomes


  • Use a variety of statistical tools to process and manage numerical data
  • Use the methods of linear regression and correlations to predict the value of a variable given certain conditions
  • Advocate the use of statistical data in making important decisions
• Statistics


  The science of collection, organizing, presenting, analyzing, and interpreting data to assist in making more effective decisions
• Divisions of Statistics


  • Descriptive statistics
  • Inferential statistics
• Descriptive statistics


  Deals with the methods of organizing, summarizing, and presenting a mass of data to yield meaningful information
• Inferential statistics


  Concerned with generalizing about a population or other groups of data based on the study of the sample
• Descriptive or Inferential statistics

  • The average life expectancy in New Zealand is 78.49 years (Descriptive)
  • A diet high in fruits and vegetables will lower blood pressure (Inferential)
  • Researchers stated that the shape of a person's ear is related to their aggression (Inferential)
  • The total amount of estimated losses from the previous hurricane was $4.2 billion (Descriptive)
• Population


  The totality of the observations with which we are concerned. It refers to a group of a total number of people, objects, or reactions that can be described as having a unique or combination of qualities. Population can be either finite or infinite
• Parameter


  Any numerical value describing a characteristic of a population and is usually represented by Greek letters
• Sample


  A finite number of objects selected from the population. It is a collection of some elements in a population or is a representative of the entire population
• Statistic


  Any numerical values describing a characteristic of a sample and usually represented by the ordinary letters of the English alphabets
• Population and Sample

  • If we consider all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter (Population)
  • If we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic (Sample)
• Population and Sample


  • A polling organization contacted 1294 teenagers who are 16 to 19 years of age and live in North America and asked whether or not they had driven a car on a highway recently. The population is the 16 to 19 year old teenagers in North America, the sample is the 1294 teenagers contacted (Population and Sample)
  • Following the election, 18% of the governors of all 50 areas of a country were female (Parameter)
  • In a national survey on substance abuse, 66.4% of respondents who were full-time college students aged 18 to 22 reported using alcohol within the past month (Statistic)
  • In a certain soccer league, 43% of the 14 teams had won more games than they had lost (Parameter)
• Sample size


  The number of respondents or subjects to form a sample
• Determining sample size


  1. For finite and known population size:

  2. For an infinite or unknown population size:

  3. Estimating a Population Mean:

  4. Estimating a Population Proportion:
• When the calculated sample size is not a whole number, it should be rounded up to the next higher whole number
• Determining sample size


  • From a population of 10,000 individuals of a certain town, the sample size needed in order to get accurate results for a certain study using a margin of error of 3% is 1067
• Random Sampling techniques


  • Simple Random Sampling
  • Systematic Random Sampling
  • Stratified Random Sampling
  • Cluster Random Sampling
  • Multi-Stage Sampling
• Simple Random Sampling


  Members from the population are selected in such a way that each individual member in the population has an equal chance of being selected. It is an equal probability sampling method (EPSEM)
• Systematic Random Sampling

  It is an equal probability sampling method (EPSEM)
• Stratified Random Sampling


  It is an equal probability sampling method (EPSEM)
• Cluster Random Sampling


  Divide the population into sections (or clusters), then randomly select some of those clusters, and then choose all members from those selected clusters
• Multi-Stage Sampling


  This method uses several stages or phases in getting random samples from the general population. This is commonly used if the research is of national scope
• Random Sampling Methods


  • Systematic random sampling: Suppose the names of 300 students of a school are sorted in the reverse alphabetical order. The management plans to choose some 15 students by starting at 5. From number 5 onwards, they will select every 15th person from the sorted list
  • Stratified random sampling: There are three bags (A, B and C), each with different balls. Bag A has 50 balls, bag B has 100 balls, and bag C has 200 balls. A man takes 5 balls from bag A, 10 balls from bag B and 20 balls from bag C
  • Simple random sampling: In an organization of 500 employees, if the HR team decides on conducting team building activities, it is highly likely that they would pick leadership roles out of a bowl
  • Cluster random sampling: An educational institution has ten branches across the country. Since the researchers can't travel to every unit to collect the required data, they decide to select three or four branches to study
  • Multi-stage sampling: The Gallup poll randomly chooses a certain number of area codes then samples a number of phone numbers from within each area code
• Non-Random Sampling techniques


  • Accidental or Haphazard or Convenience sampling
  • Purposive sampling
• Accidental or Haphazard or Convenience sampling

  Methods done are normally biased since the researcher considers his/her convenience in the collection of the data
• Purposive sampling


  It is based on certain criteria laid down by the researcher. People who satisfy the criteria are interviewed
• Sub-categories of Purposive sampling

  • Modal instance sampling
  • Heterogeneous or maximum variation sampling
  • Homogeneous sampling
  • Critical case sampling
  • Typical case sampling
• Sampling techniques


  • Systematic random sampling
  • Stratified random sampling
  • Simple random sampling
  • Cluster random sampling
  • Multi-stage sampling
• Non-random sampling techniques


  • Accidental or Haphazard or Convenience sampling
  • Purposive sampling
• Purposive sampling


  It is based on certain criteria laid down by the researcher. People who satisfy the criteria are interviewed.
• Sub-categories of purposive sampling

  • Modal instance sampling
  • Expert sampling
  • Quota sampling
  • Snowball sampling
• Modal instance sampling


  Sampling the most frequent cases. The problem is identifying the "modal" case.
• Expert sampling


  Involves assembling a sample of persons with known or demonstrable experience and expertise in some area.
• Quota sampling


  Selecting items non-randomly according to some fixed quota.
• Snowball sampling


  Begin by identifying someone who meets the criteria for inclusion in your study. You ask them to recommend others who they may know who also meet the criteria.
• Types of statistical data

  • Qualitative (Categorical) data
  • Quantitative (Numerical) data
• Qualitative (Categorical) data


  Generally described by words or letters
• Sub-types of qualitative data

  • Dichotomic
  • Polynomic
• Dichotomic


  Takes the form of a word with two options, such as gender - male or female.
• Polynomic


  Takes the form of a word with more than two options, such as education - primary school, secondary school and university.