Inferential Statistics

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Inferential Statistics
Random Sampling Laws of Probability Sampling
Distribution Sampling Fluctuation Central Limit
Theorem Standard Error Confidence Levels/Intervals
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Introduction to Hypothesis Testing

• a decision-making process for evaluating claims
  about a population

Inferences Based on a Single Sample
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Hypotheses - two kinds

• Research hypotheses
• Statistical hypotheses

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Research hypotheses are the ones that are stated
in relatively plain English about what you think
will be the outcome of the research.
The Scientific Method
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Science is best defined as a careful,
disciplined, logical search for knowledge about
any and all aspects of the universe, obtained by
examination of the best available evidence and
always subject to correction and improvement
upon discovery of better evidence.
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The scientific method is the best way yet
discovered for separating the truth from lies and
delusion

• Observe some aspect of the universe.
• Invent a tentative description, called a
  hypothesis, that is consistent with what you have
  observed.
• Use the hypothesis to make predictions.
• Test those predictions by experiments or further
  observations and modify the hypothesis in the
  light of your results.
• Repeat steps 3 and 4 until there are no
  discrepancies between theory and experiment
  and/or observation.

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When consistency is obtained the hypothesis
becomes a theory and provides a coherent set of
propositions which explain a class of
phenomena. A theory is then a framework within
which observations are explained and predictions
are made.
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The results obtained using the scientific method
are repeatable

• The great advantage of the scientific method is
  that it is unprejudiced one does not have to
  believe a given researcher, one can redo the
  experiment and determine whether his/her results
  are true or false.
• The conclusions will hold irrespective of the
  state of mind, or the religious persuasion, or
  the state of consciousness of the investigator
  and/or the subject of the investigation.

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Research Hypothesis Statements reference expected
outcomes Increasing solid ink density produces
a longer tonal range. Serifs on type increase
reading comprehension.
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Most commonly, hypotheses take three formats
a question Does temperature affect ink flow?
a conditional statement Temperature may affect
ink flow. an If, then statement If ink
viscosity is related to temperature, then
increasing the temperature will increase ink
flow.
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Statistical Hypotheses are probabilistic
mathematical statements concerning population
values, stated in terms of the parameters used in
the research.
A parameter is a population mean, proportion,
variance Must be statedbefore analysis
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Hypothesis testing

• Is a formal way of testing claims such as these
  and is closely related to confidence intervals.

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Confidence Interval

• By computing a confidence interval, we can obtain
  a range of likely values for the population
  parameter we're estimating
• Not only that, but we can do a test to see if
  claims were correct by seeing if the confidence
  interval captured the claimed value

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Thus, for a 95 confidence interval.. If we
take a large number of sample means, 95 of the
time the distance between an individual sample
mean and the population mean will be less than
1.96 standard deviations of the sampling
distribution
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Example

• a manufacturer claims that the average lifetime
  of an electronic component is 32 hours.
• We could take a sample of electronic components
  of size n and measure their lifetime. By
  measuring the sample mean and standard error, we
  can compute a 95 confidence interval.
• If 32 fell within the interval, we would believe
  the claim of the manufacturer. If it didn't fall
  within the interval, we wouldn't believe the
  claim

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Each study has two Statistical Hypotheses

• Null hypothesis, H0
• A statistical hypothesis stating that there is no
  difference between a parameter and a specific
  value
• or that there is no difference between 2
  parameters.

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Each study has two Statistical Hypotheses

• Alternative hypothesis, Ha or H1
• A statistical hypothesis that states a specific
  difference between a parameter and a specific
  value
• or specifies that there is a difference between
  2 parameters.

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Basic Idea

• If the sample mean looks as though it could have
  come from the sampling distribution given by the
  null hypothesis, then we will accept the null
  hypothesis.
• If the sample mean is way out on the tail, or
  completely outside the sampling distribution
  given by the null hypothesis, we should reject
  the null hypothesis.
• Only work we have to do decide what is inside,
  and what is outside the distribution!
• Have to
  DRAW THE LINE!

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Sampling Distribution of a Mean
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• In hypothesis testing, the researcher must
• define the population under study
• state the particular hypotheses that will be
  investigated
• give the significance level
• select a sample from the population
• collect the data
• perform the calculations required for the
  statistical test
• and reach a conclusion.

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Hypothesis testing in science is a lot like the
criminal court system in the United States. How
do we decide guilt?

• Assume innocence until proven guilty.
• Evidence is presented at a trial.
• Proof has to be beyond a reasonable doubt.

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A jury's possible decision

• guilty
• not guilty

Note that a jury cannot declare somebody
innocent just not guilty
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Do juries ever make mistakes?

• If a person is really innocent, but the jury
  decides they are guilty, then they've sent an
  innocent person to jail.
• Type I error.
• If a person is really guilty, but the jury finds
  them not guilty, a criminal is walking free on
  the streets.
• Type II error.

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In our criminal court system, a Type I error is
considered more important than a Type II error,
so we protect against a Type I error to the
detriment of a Type II error. This is the same
as in statistics.
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Null and Alternative Hypotheses

• Science, in general, operates by disproving
  unsatisfactory hypotheses and proposing
  new-and-improved hypotheses which are testable.
• The approach taken in statistics is exactly this
  scientific method.
• We start with a hypothesis which we assume is
  correct. We call this the null hypothesis or H0,
  and our goal is to reject H0 in favor of the
  alternative hypothesis, H1.

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Null hypotheses are denoted H0 while
alternative hypotheses can be denoted as either
H1 or Ha Example Null hypothesis ------------
H0 m1 m2 Alternative hypothesis ----- H1
m1 ltgt m2
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Testing Statistical Hypotheses

• Assume a claim is true (status-quo). Call this
  claim the null hypothesis, .
• Look at the evidence (which we've collected from
  a sample) to make our decision.
• Goal prove beyond a reasonable doubt'' that
  the null hypothesis is false.
• Our possible decisions are
• reject H0 in favor of H1
• fail to reject H0

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Example

• A national magazine claims that the average
  college student watches less television than the
  general public.
• The national average is 29.4 hours per week, with
  a standard deviation of 2 hours.
• A researcher samples 25 college students,
  finding the mean27. Is there evidence to support
  the magazines claim?

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Note that Ha is the hypothesis on which the
burden of proof is placed. this is the claim
the researcher is investigating or the hypothesis
proposed by the experimenter.
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Once H0 and Ha have been formulated and the
experiment has been run, we either reject H0 or
fail to reject H0. We may be correct or
incorrect in our decision about H0. There are 4
possibilities
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Type I and Type II Errors

• Type I error a
• We reject H0 when it is really true.
• Type II error b
• We fail to reject H0 when Ha is really true.

A Type I error is considered more serious.
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Type I and Type II Errors
Correct decisions a) Reject H0 and Ha is
really true. b) Fail to reject H0 and H0
is really true.
It is important to emphasize that we can either
reject or fail to reject .. in the same sense,
a jury can only find someone guilty or not
guilty, but not innocent.
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The decision to reject or not reject the null
hypothesis does not prove anything this could
only be done by using the entire population. The
decision, then, is made on the basis of
probabilities. When there is a large difference
between the sample statistic and the hypothesized
parameter value, the null hypothesis is probably
not true.
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Level of Significance

• To determine how large a difference is necessary
  to reject H0, the level of significance is used.
• This is the maximum probability of committing a
  Type I error and is symbolized by alpha a

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a Level of Significance

• 1. Determines how far in to draw the line
• Determines the rejection and acceptance regions
• 2. Selected by researcher at start
• Typical values are .01, .05

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Level of Significance is a Probability

• Alpha is the probability
• of making a Type I Error
• (reject H0 when it is really true)
• and is called the level of significance or the a
  level.
• 1 - a is called the level of confidence.

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The researcher decides what level of significance
to use depending on the seriousness of the Type I
error. Usually, an a .05 or an a .01 is
used a 5 or 1 chance of rejecting a true null
hypothesis. After the significance level is
chosen, an appropriate statistical test (and
accompanying table) is chosen. The statistical
test is used to calculate the test statistic, and
the table is used to get the critical value.
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Critical Value

• Separates the critical region from the
  non-critical region.

Z a/2
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Critical Region The region of values of the test
statistic that indicates there is a significant
difference and H0 should be rejected. Non-critica
l region The region of values of the test
statistic that indicates that the difference was
probably due to chance and that H0 should not be
rejected.
Also known as Probability Regions
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Test Statistic

• A value that is calculated and compared to the
  critical value in order to make a conclusion
  about whether to reject the null hypothesis or to
  fail to reject the null hypothesis.

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Choosing an Alpha Level - Researchers select
the alpha level they wish to use.

• Make a too large and you will commit too many
  Type I errors.
• Make a too small and you will not detect true
  effects when they exist.

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The Meaning of Statistical Significance

• A finding is described as statistically
  significant, when it can be demonstrated that the
  probability of obtaining such a result by chance
  only, is relatively low.
• It means that the observed result is unlikely to
  occur by chance alone.
• It means that the results are reliable and likely
  to be repeatable.

It does not mean that the effect is large,
important or meaningful.
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One-tail vs. Two-tail Tests
One-tail test have a single rejection One-tail
test should only be done when Theory makes a
directional prediction. There is strong
empirical evidence of direction differences H0
m1 gt m2         H1 m1 lt m2 or H0 m1 lt m2  
      H1 m1 gt m2
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Rejection Region for One-Tail Test
Sampling Distribution
Level of Confidence
1 - ?
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One-tail vs. Two-tail Tests
Two-tail tests have two rejection regions. H0
m1 m2         H1 m1 ltorgt m2
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Rejection Regions (Two-Tailed Test)
Sampling Distribution
1 - ?
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Critical Values

• Critical values of a statistical indicate the
  beginning of the rejection regions.
• Among all the sets of possible values, we must
  choose one that we think represents the most
  extreme evidence against the hypothesis. That is
  called the critical region of the test statistic.
  
• The probability of the test statistic falling in
  the critical region when the hypothesis is
  correct is called the alpha value of the test

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If the test statistic is inside the critical
region, then our conclusion is either The
hypothesis is incorrect or An event of
probability less than or equal to alpha has
occurred. The researcher has to choose
between these logical alternatives.
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If the test statistic is outside the critical
region, the only conclusion is that There is
not enough evidence to reject the hypothesis.
This is not the same as evidence for the
hypothesis. That evidence we cannot
obtain. Statistical research progresses by
eliminating error, not by finding the truth.
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Review Hypothesis Testing Facts

• Hypotheses
• Null Hypothesis
• The accepted explanation, status quo. This is
  what we're trying to disprove.
• Alternate Hypothesis
• What the researcher or scientist thinks might
  really be going on, a (possibly) better
  explanation than the null.

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Review Hypothesis Testing Facts

• Test
• The goal of the test is to reject H0 in favor of
  H1 . We do this by calculating a test statistic
  and comparing its value with a value from a
  statistical table, the critical value.
• If our test statistic is more extreme than our
  critical value, then it falls within the
  rejection region of our test and we reject H0. We
  can set up the rejection region before computing
  our test statistic.
• Decisions
• Reject H0.
• Fail to reject H0.
• Errors
• Type I/ Reject H0 when H0 is really true.
• Type II/ Fail to reject H0 when H0 is really
  false.

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H0 Testing Steps

• 1. State H0, H1, ?, and n
• 2. Collect data
• 3. Compute test statistic
• 4. Identify rejection/acceptance regions
• 5. Draw conclusions.

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One Population Tests