Download Inferential Statistics - Indiana University Bloomington

S519: Evaluation of
Information Systems
Social Statistics
Inferential Statistics
Chapter 8: Significantly
significant

The “4 Steps” of Hypothesis Testing:
1.
2.
3.
4.
State the hypothesis
Set decision criteria
Collect data and compute sample statistic
Make a decision (accept/reject)
This week
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Types of variables
Central Limit Theorem
Steps of hypothesis testing
Type I and Type II errors
Types of Variables (Betweensubjects vs. Within-subjects)
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Between-subjects variable: a characteristic
that varies between subjects (e.g. age,
gender etc.).
Within-subjects variable: a characteristic of
individuals that varies with time. (the same
individuals are compared at different points in
time).
Types of Variables
(independent vs. dependent )

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The independent variable is the one that is
manipulated.
The dependent variable is the one that is
observed.
Example I

A sport psychologist is interested in the
relationship between reach and knockouts in
boxers. She measures the reach of 20 pro
boxers, and then observes the number of
knockouts each scores over one year of
fighting.

Between-subjects or within-subjects?
Example II

A researcher is testing the effect of alcohol
on memory performance. He gives one group
of subjects a bottle of vodka, and another a
nonalcoholic substance that tastes like
vodka. Each group then learns a list of words,
and attempts to recall them. Number of words
correctly recalled for each group is recorded.

Between-subjects or within-subjects?
What are the independent and dependent variables?
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Example III
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A social psychologist is interested in gender
differences in math performance. She
randomly selects men and women from IU
and has them solve a series of equations.
Number of equations correctly solved for
each participant is recorded.

Between-subjects or within-subjects?
What are the independent and dependent variables?
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A fallacy
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In a survey, one of the questions asks
respondents whether they had breakfast
today, another asks them whether they are
satisfied with their work today.
You found that 80% of the respondents who
answered yes on the breakfast question said
they were satisfied with their work.
You concluded that eating breakfast will
make people feel satisfied about their work.
Justified or not?
A fallacy

Alternative I (between-subjects)
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Alternative II (within-subjects)
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Two groups (same number of people, similar backgrounds,
randomly chosen)
One have breakfast; one without having breakfast
Test their satisfaction towards work
One group
Having breakfast the first day; without having breakfast the
second day
Test their satisfaction towards work
Still, there are confounding factors (weather, order, etc.)
Distribution of Sample Means
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So far, we’ve talked about samples of size 1.
In an experiment, we take a sample of several
observations and try to make generalizations
back to the population.
How do we estimate how good a representation
of the population the sample we obtain is?
We can rely on the assumption that most
populations are normally distributed, and apply
the Central Limits Theorem.
Sampling Distributions

The distribution of sample means contains all
sample means of a size n that can be
obtained from a population.
Central Limit Theorem

For any population with mean μ and standard
deviation σ, the distribution of sample means
for sample size n will have a mean of μ and a
standard deviation of  / n , and will approach
a normal distribution.
Central Limit Theorem

The Central Limit Theorem tells us that for
any DSM of samples of size n:
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μM =μ
M 

n
Central Limit Theorem
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The mean of the distribution of sample means is
called the expected value of M.
The standard deviation of the distribution of
sample means is called the standard error of M.
standard error = σM
Standard deviation: standard distance between
a score X and the population mean μ.
Standard error: standard distance between a
sample mean M and the population mean μ.
Probability and the Sample
Means Distribution
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SAT-scores (μ=500, σ=100).
What is the probability (M>540)?
If
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Choose one randomly from the population;
Take sample n=25.
Step 1: State Hypothesis

Hypothesis is stated in terms of a population
parameter (e.g. μ): assume null unless
sufficient evidence to reject it

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Null Hypothesis: no change in population
parameter, treatment has no effect. H0
Alternate (Scientific) Hypothesis: opposite of H0,
treatment has an effect. H1 (directional or non
directional)
Step 2: Set Criteria

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By selecting a probability value (level of
significance or alpha level), e.g. α = 0.05, 0.01,
0.001
Critical region is region of extreme sample
values (unlikely to be obtained if H0 is true). If
sample data fall within the critical region, H0 is
rejected.
Determine exact values for boundaries of critical
region by using α and unit normal table.
Step 2: Set Criteria
Step 3: Collect Data/Statistics
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Select random sample and perform
“experiment”.
Compute sample statistic, e.g. sample mean.
Locate sample statistic within hypothesized
distribution (use z-score).
Is sample statistic located within the critical
region?
Step 4: Decision
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Possibility: sample statistic is within critical
region. Reject H0.
Possibility: sample statistic is not within
critical region. Do not reject H0.
Hypothesis Testing: An
Example
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It is known that corn in Bloomington grows to an average
height of μ=72 σ=6 six months after being planted.
We are studying the effect of “Plant Food 6000” on corn
growth. We randomly select a sample of 40 seeds from
the above population and plant them, using PF-6000
each week for six months. At the end of the six month
period, our sample has a height of M=78 inches. Go
through the steps of hypothesis testing and draw a
conclusion about PF-6000
1. State hypotheses; 2. Chance model/critical region;
3. Collect data; 4. Decision and conclusion
Step 1: State Hypotheses

In words
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Null: PF6000 will not have an effect on corn
growth
Alt: PF6000 will have an effect on corn growth
In “code” symbols:
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H0 :μ = 72
H1 :μ ≠ 72
Step 2: Chance Model and
Critical Values
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a) Distribution of Sample Means:
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b) Set alpha level α=.05 ∴ zcrit = ±1.96
Step 2: Chance Model and
Critical Values
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c) Compute critical values to correspond to
zcrit
Step 3: Do Experiment

For the question, this part has already been
done for us, we just need to compare this
obtained sample mean to our chance model
to determine if any discrepancy between our
sample and the original population is due to:

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1. Sampling Error
2. A true effect of our manipulation
Step 4: Decision and
Conclusion
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Mcrit is 70.14 (lower) or 73.86 (upper)
If Mobt exceeds either of these critical values
(i.e., is out of the “chance” range, we reject
H0. Otherwise, cannot reject H0)
Mobt = 78 Mcrit = 73.86
Mobt exceeds Mcrit ∴ Reject H0
Conclusion: We must reject the null
hypothesis that the chemical does not
produce a difference. Conclude that PF6000
has an effect on corn growth.
Uncertainty and Errors
Accept the null hypothesis Reject the null hypothesis
The null
hypothesis is
really true
, you accepted a null when
it is true
[there is really no difference
between the groups]
Type I error: reject a null
hypothesis when it is true
(represented by the Greek
letter alpha, α)
[there is really no difference
between the groups]
The null
hypothesis is
really false
Type II error: accepted a
false null hypothesis
(represented by Greek letter
beta, β)
[there really are differences
between the two groups]
, you rejected the null
hypothesis which is false.
[there really are differences
between the two groups]
Critical regions for different
values of α
A template for significant test
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1. a statement of the null hypothesis
2. setting a level of risk associated with the null hypothesis (level of
significance or Type I error, p)
3. select a proper statistical test (see Fig 8.1)
4. set up the sample and experiment, and compute the test statistic
value
5. determine the value needed for rejection of the null hypothesis
using proper tables – critical value (see appendix)
6. compare the computed value and the obtained value
7. if computed value > critical value: reject the null;
if computed value < critical value: accept the null