# Hypothesis testing and statistical inference

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```					Hypothesis testing and
statistical inference

Research Process and Design
Spring 2006
Class #10 (Week 11)
Today’s objectives

   To review elements of last weeks class
   To understand logic of inferential statistics
   To explore hypothesis testing
   To get feedback on first draft of methods
section

Research Process and Design (Umbach)   2
Logic of Inferential Statistics

   Hope to say something with confidence about
the population based on a sample
   Use probabilities to state the degree of
confidence

Research Process and Design (Umbach)   3
Probability samples and standard errors
   Random selection
   Human influence is removed from the selection
process
   E.g., dice, random number generator
   Probability samples use random selection to
draw a subset of the sampling frame
   Sampling error arises because of this
   Standard errors allow us to quantify this error

Research Process and Design (Umbach)   4
Laws of Sampling Theory
   Whenever a random sample is taken from a
population there will be sampling error.
   If sample is truly random, then
characteristics of sample will be an unbiased
estimate of population characteristics.
   As sample size increases, the range (the
size) of sampling error decreases.

Research Process and Design (Umbach)   5
Central Limit Theorem
   The sampling distribution, or the distribution of the
sampling error for any sample drawn from a given
population, approximates a normal curve.
   Standard error - standard deviation of the sample
estimates of means that would be formed if an infinite
number of samples.

s
se =
n

Research Process and Design (Umbach)   6
Standard error
   Relies on the concept of repeated samples
from a population
   Due to chance, the means of these samples will
vary around the population mean
   We can measure this variance and determine how
much the typical sample will deviate from the
population mean (i.e., the standard deviation or
SD)
   This SD is the standard error (SE)
   http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

Research Process and Design (Umbach)           7
Standard errors

s
   Standard error of the mean: s X 
n
   s is the SD from our sample; n is sample size
   We can see that as n increases, SE decreases
   Different formulas for different statistics
(proportions, comparing two means, etc.), but they
have a similar form

Research Process and Design (Umbach)   8
Confidence Intervals

   The range within which the parameter in
question could be expected to be included a
specified percentage of the time if procedure
were to be repeated.
C(se)
C = Z statistic associated with the confidence level;
1.96 corresponds to the .95%,
2.33 corresponds to the 98% level,
and 2.58 corresponds to the 99% confidence level

Research Process and Design (Umbach)   9
Standard errors

   Confidence intervals (CI) use SE and tell us
the precision of our estimates
   95% CI for a mean = X  1.96*SE
   Very specific definition: if we calculated similar CIs
on 100 similar samples, 95% of them would
bracket the population parameter
   Does not mean there is a 95% probability that
population parameter falls in your CI – either it
does or it doesn’t
   http://www.ruf.rice.edu/~lane/stat_sim/conf_interval/

Research Process and Design (Umbach)   10
Standard errors

   Margin of error in polls is a confidence
interval, usually a 95% CI

Research Process and Design (Umbach)   11
Central Limit Theorem
   The sampling distribution, or the distribution
of the sampling error for any sample drawn
from a given population, approximates a
normal curve.
   Standard error - standard deviation of the
sample estimates of means that would be
formed if an infinite number of samples is
known as the standard error.
s
se =
n

Research Process and Design (Umbach)   12
Confidence Intervals and Hypothesis
Testing
   The range within which the parameter in
question could be expected to be included a
specified percentage of the time if procedure
were to be repeated.
C(se)
C = Z statistic associated with the confidence level;
1.96 corresponds to the .95%,
2.33 corresponds to the 98% level,
and 2.58 corresponds to the 99% confidence level

Research Process and Design (Umbach)   13
Normal Curve

   Symmetric around mean

   Skewness
   Positively skewed
   Negatively skewed

   Kurtosis
   Leptokurtic
   Platykurtic

Research Process and Design (Umbach)   14
Research Process and Design (Umbach)   15
Hypotheses

   How do we know that differences between
groups are not due to sampling error? How
big does a correlation need to be to know that
difference is real and not a result of sampling
error?
   Hypothesis testing?
   How do we develop hypotheses?
   Null hypothesis
   Research (or alternative) hypothesis

Research Process and Design (Umbach)   16
Statistical Hypotheses—Null
Hypothesis (H0)

   Statistical statement of no difference
in population
   No difference between performance
of a group and accepted benchmark
   e.g., passing a test

Research Process and Design (Umbach)   17
Statistical Hypotheses—Null
Hypothesis (H0)
   No difference between two or more groups
that are being compared
   e.g., experimental and comparison groups
   No relationship between or among predictor
and criterion variables
   e.g., relationship between attitude and
achievement does not exist; it is not different from
zero

Research Process and Design (Umbach)   18
Statistical Hypotheses--Alternative
hypothesis (H1)
   Statistical statement that includes all results not explicitly stated
in the null hypothesis
   Reflects the existence of differences or relationships
 The performances of students in the experimental group exceeds
the performances of those in the comparison group
 A relationship between attitude and achievement does exist

   Usually reflects the research hypothesis

Research Process and Design (Umbach)          19
Choice of probability level

   Rejection region or significance level
   p<.10,p<.05, p<.01, p<.001
   Determined a priori
   Significance
   Interpreting results
   One-tailed test
   Two-tailed test

Research Process and Design (Umbach)   20
Error in hypothesis testing
State of Nature
Null hypothesis                        Null hypothesis
is true                               is false

Reject null
Type I error                        Correct decision
hypothesis

Fail to reject
Correct decision                         Type II error
null hypothesis

Research Process and Design (Umbach)                     21
Feedback on the method section
 Do you have all of the information you need
to fully assess the method? What do you
need?
 Is the study feasible as described in the
method?
 How is the study (1)Maximizing experimental
variation, (2)Minimizing Error Variation, and
(3) Controlling Extraneous (Confounding)
Variation
Don’t hesitate to ask the other group for help in
areas where you are having trouble.

Research Process and Design (Umbach)   22
In two weeks…
   Analysis of variance (ANOVA)

   Reading for next week:
   Jaeger – Chapters 12-14

   Assignment due: Draft of proposal. Bring 3
copies to class.

   NO CLASS NEXT WEEK. Work on proposals.

Research Process and Design (Umbach)   23

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