Introduction to Math 200 Elementary Statistics

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Introduction to Math 200 Elementary Statistics

• Uncertainty (Variability) Exists
• Benjamin Franklin (1706-1790) said,
• nothing is certain in life but death and taxes.
• Is time of death known?
• Is amount owed in taxes?
• Uncertainty (Variability) is Unavoidable
• reading from Denny Gaines

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Uncertainty Exists

• James 413-15
• 13 Now listen, you who say, "Today or
  tomorrow we will go to this or that city, spend a
  year there, carry on business and make money. 14
  Why, you do not even know what will happen
  tomorrow. What is your life? You are a mist that
  appears for a little while and then vanishes. 15
  Instead, you ought to say, "If it is the Lord's
  will, we will live and do this or that.

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Uncertainty in Nature

• Location of atomic particles
• Weather
• Diffusion of Molecules
• Drift of Genes in a Population
• Spread of Disease
• Failure Time

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Variability Exists

• Results vary in repeated experiments under
  identical conditions
• Instruments are limited to finite precision.
  Calculations made on measurements with finite
  precision will always have an associated error.

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Disorder in Nature

• Spider Silk
• Is it rubber or steel?
• 4X stronger than steel
• Absorbs 3X more energy than Kevlar
• Strength comes from order in the structure of
  amino acids (crystalline polymer).
• Extensibility comes from molecular disorder
  (randomness) in the arrangement of amorphous
  protein chains connecting the crystals.

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Uncertainty, Variability, Disorder

• A negative notion of these ideas existed among
  scientists for many years
• Uncertainty viewed as necessary evil that could
  be tamed by clever ideas and (as a last resort)
  inferential statistics.
• Variability and disorder seemed to be in direct
  conflict with predictability.
• The presence of uncertainty and disorder in
  nature was often thought to be holding back
  science and the discovery of truth.

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Changing Scientific View of Chance

• Albert Einstein said about the notion of
  chance in nature
• God does not play dice.

• Niels Bohr responded,
• Stop telling God what to do.

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Chance in Science

• A chance event is one (not caused by the organism
  itself) which could not have been predicted even
  with knowledge of the initial conditions
• Chance is not a causal agency which makes things
  happen, but rather a description of the way we
  understand the workings of certain events in the
  world around us.

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Christian View of Chance

• Many Christians are still uncomfortable
  considering the notion that things in the world
  can happen by chance.
• Partly due to the presence of chance in current
  scientific theories like evolution
• God uses chance and disorder in his design of
  nature.

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Nature Declares God

• Romans 118-20
• 18 The wrath of God is being revealed from
  heaven against all the godlessness and wickedness
  of men who suppress the truth by their
  wickedness, 19 since what may be known about God
  is plain to them, because God has made it plain
  to them. 20 For since the creation of the world
  God's invisible qualitieshis eternal power and
  divine naturehave been clearly seen, being
  understood from what has been made, so that men
  are without excuse.

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• Blaise Pascal (1623-1662)
• In faith there is enough light for those who
  want to believe and enough shadows to blind those
  who don't.

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Molecular Diffusion

• Transports molecules solely by the random motions
  of other molecules
• At room temperature, each oxygen molecule
  collides with a neighbor about 6 billion times
  per second.
• It is impossible to predict the location of a
  single molecule even a fraction of a second ahead
  of time.

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Designed Uncertainty in Biology

• Gods design uses molecular diffusion to send
  signals across the junction between vertebrate
  motor nerves and muscles.
• No direct transmission is necessary!!

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Christian View of Chance

• How can God can operate in a world where things
  sometimes seem to happen by chance?
• Events that appear random from a human
  perspective need not be uncaused from a divine
  perspective.
• God can determine the outcome of what man calls
  random events.

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Christian View of Chance

• God created the world and set in place what man
  calls laws of nature
• An example is the law of gravity
• Some laws of nature are now thought to involve
  chance events
• An example is genetics

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God Does Play Dice

• God built uncertainty into the world around us in
  many amazing ways.
• What seems like uncertainty to us is in fact
  certain to God.
• Proverbs 1633
• The lot is cast into the lap, but its every
  decision is from the LORD.

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Gods People Cast Lots

• Acts 123-26
• 23 So they nominated two men Joseph called
  Barsabbas (also known as Justus) and Matthias.
• 24 Then they all prayed for the right man to be
  chosen. "O Lord," they said, "you know every
  heart. Show us which of these men you have chosen
• 25 as an apostle to replace Judas the traitor in
  this ministry, for he has deserted us and gone
  where he belongs."
• 26Then they cast lots, and in this way Matthias
  was chosen and became an apostle with the other
  eleven.

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Probability Statistics

• Probability
• Called the laws of chance by many
• Useful for modeling uncertainty
• Statistics
• Based on principles of probability
• Useful for making decisions in the face of
  uncertainty

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Statistics

• The science of collecting, simplifying, and
  describing data, as well as making inference
  based on the analysis of data.
• Descriptive Statistics
• Simplifying, summarizing, describing, etc.
• Inferential Statistics
• Making inference, drawing conclusions

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Statistics is a Process

• Statistics is the process of making sense of
  data.
• Gathering Data - a critical step
• Summarizing Data descriptive statistics
• Analyzing Data inferential statistics
• Communicating Results - a clear statement of the
  proper interpretation is important

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Why Study Statistics?

• One obvious reason is that is required!!
• A basic understanding is becoming
  expected/necessary in todays world.
• Many jobs require it.
• Ignorance of statistics can be used against you.
• It has application in every field.

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Population versus Sample

• You must be able to distinguish between a
  population and a sample.
• Population
• The set of all possible outcomes.
• Sample
• A subset of the possible outcomes.
• Purpose is to accurately reflect the population.
• A large sample size does not guarantee a good
  sample.

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Parameter versus Statistic

• You must be able to distinguish between a
  parameter and a statistic.
• Parameter
• A numerical property of the population.
• A fixed value.
• Statistic
• A numerical property of the sample.
• A random value.

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Overview Statement

• Descriptive statistics from a sample can be used
  to draw conclusions about a population
  (parameter).

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Types of Data

• Qualitative Data (Categorical)
• Nominal Data
• No natural Ordering.
• Ordinal Data
• Natural ordering exists.

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• Quantitative Data (Numerical)
• Discrete Data
• Finite (countable) number of outcomes possible.
• Typically counting something.
• Continuous Data
• Infinite number of outcomes possible
  corresponding to an interval on the number line.
• Typically measuring something.
• Interval Scale no meaningful zero
• Ratio Scale included a meaningful zero

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Gathering Data

• A critical step in the process of making sense
  out of data.
• Specify the problem to make sure you understand
  it
• Identify potentially significant variables and
  factors
• Choose an appropriate design.
• Collect data.

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Sample Survey

• Passive tool for collecting data.
• Sampling Techniques
• Simple Random Sample equally likely outcomes
• Stratified Random Sample equally likely
  outcomes within strata (sub-populations)
• Cluster Sample equally likely outcomes from a
  subset of strata
• Systematic Sample randomly select first sample
  and system determines the remaining samples
• Convenience Sample not random

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Sample Survey

• Data Collection Techniques
• Personal Interview
• Telephone Interview
• Self-Administered Interview
• Direct Observation
• The outcome of a survey can be dramatically
  changes by the design of a questionnaire.

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Experimental Study

• An active tool for collecting data that involves
  manipulation of variables and observation of the
  effect on other variables.
• Experimental Design
• Completely Randomized Design (CRD)
• Randomized Block Design (RBD)
• Blocking is used to reduce or eliminate the bias
  or variability due to a known factor.

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Observational Study

• A study with limited control over the conditions
  of the experiment.