Transcript Document

Education Research
250:205
Writing Chapter 3
Objectives





Subjects
Instrumentation
Procedures
Experimental Design
Statistical Analysis
 Displaying data
 Analyzing data
 Descriptive statistics
 Derived scores
 Inferential statistics




Introduction
Confidence intervals
Comparison of means
Correlation and regression
Introduction

Statistical inference: A statistical process
using probability and information about a
sample to draw conclusions about a
population and how likely it is that the
conclusion could have been obtained by
chance
Distribution of Sample Means

Assume you took an infinite number of
samples from a population
 What
would you expect to happen?
Assume a population consists of 4 scores (2, 4, 6, 8)
Collect an infinite number of samples (n=2)
Total possible outcomes: 16
p(2) = 1/16 = 6.25%
p(3) = 2/16 = 12.5%
p(4) = 3/16 = 18.75%
p(5) = 4/16 = 25%
p(6) = 3/16 = 18.75%
p(7) = 2/16 = 12.5%
p(8) = 1/16 = 6.25%
Central Limit Theorem

The CLT describes ANY sampling
distribution in regards to:
Shape
2. Central Tendency
3. Variability
1.
Central Limit Theorem: Shape
All sampling distributions tend to be
normal
 Sampling distributions are normal when:

 The
population is normal or,
 Sample size (n) is large (>30)
Central Limit Theorem: Central Tendency

The average value of all possible sample
means is EXACTLY EQUAL to the true
population mean
µ = 2+4+6+8 / 4
µ=5
µM = 2+3+3+4+4+4+5+5+5+5+6+6+6+7+7+8 / 16
µM = 80 / 16 = 5
Central Limit Theorem: Variability

The standard deviation of all sample means
is = SEM/√n
 Also
known as the STANDARD
ERROR of the MEAN (SEM)
Central Limit Theorem: Variability

SEM
Measures how well statistic estimates
the parameter
The amount of sampling error that is
reasonable to expect by chance
Central Limit Theorem: Variability

SEM decreases when:
 decreases
 Sample size increases
 Population

SEM
= /√n
Other properties:
 When
 As
n=1, SEM
= population SD
SEM decreases the sampling distribution
“tightens”
So What?
A sampling distribution is NORMAL and
represents ALL POSSIBLE sampling
outcomes
 Therefore PROBABILITY QUESTIONS
can be answered about the sample
relative to the population

Introduction

1.
2.
Two main categories of inferential
statistics
Parametric
Nonparametric
Introduction
Parametric or nonparametric?
 What is the scale of measurement?

or ordinal  Nonparametric
 Interval or ratio  Answer next question
 Nominal

Is the distribution normal?
 Parametric
 No  Nonparametric
 Yes
Objectives





Subjects
Instrumentation
Procedures
Experimental Design
Statistical Analysis
 Displaying data
 Analyzing data
 Descriptive statistics
 Derived scores
 Inferential statistics




Introduction
Confidence intervals
Comparison of means
Correlation and regression
Confidence Intervals
Application: Estimation of an unknown
variable that is unable or undesirable to be
measured directly
 Confidence intervals estimate with a
certain amount of confidence

Confidence Intervals

1.
Components of a confidence interval:
The level of confidence
-Chosen by researcher
-Typically 95%
-What does it mean?
2.
3.
The estimator (point estimate)
The margin of error
X% CI = Estimator +/- Margin of error
Confidence Intervals: Example



A researcher is interested in the amount of $
budgeted for special education by elementary
schools in Iowa
Select a random sample from the population and
collect appropriate data
Results:
average $ spent was $56,789 (95% CI: $51,111 –
62,467)
 The average$ spent was $56,789 +/- 5,678 (95% CI)
 The
Objectives





Subjects
Instrumentation
Procedures
Experimental Design
Statistical Analysis
 Displaying data
 Analyzing data
 Descriptive statistics
 Derived scores
 Inferential statistics




Introduction
Confidence intervals
Comparison of means
Correlation and regression
Comparing Means  Hypothesis Tests

Compare two means
 Compare
 Compare
 Compare

Compare three or more means
 Compare
 Compare

means between groups
means within groups
Compare means as a function of two or more
factors (independent variables)
 Factorial

a mean two a known value
means between groups
means within groups
designs
Compare means of multiple dependent variables
 Multivariate
designs
Hypothesis Tests – A Step by Step
Process
Step 1: State the null hypothesis
 Step 2: Select level of significance
 Step 3: Sample data
 Step 4: Choose statistic
 Step 5: Calculate the statistic
 Step 6: Interpret the statistic

Step 1: Null Hypothesis



Recall  Null hypothesis is a statement of no
effect
The test statistic either accepts or rejects the H0
Create H0 for following tests:
 Are
females in Iowa taller than 6 feet?
 Do 6th grade boys score differently than 6th grade
females on math tests?
 Does an 8-week reading program affect reading
comprehension in 3rd graders?
Step 1: Null Hypothesis
The statistic will “test” the H0 based on
data
 No statistic is perfect  The probability of
error always exists
 There are two types of error:

I error  Reject a true H0
 Type II error  Accept a false H0
 Type
Step 1: Null Hypothesis
How does one control for
Type I and II error?
Researcher
Conclusion
Accept H0
Reality No real difference
About exists
Test
Real difference
exists
Reject H0
Correct
Type I
Conclusion error
Type II
error
Correct
Conclusion
Hypothesis Tests – A Step by Step
Process
Step 1: State the null hypothesis
 Step 2: Select level of significance
 Step 3: Sample data
 Step 4: Choose statistic
 Step 5: Calculate the statistic
 Step 6: Interpret the statistic

Step 2: Significance Level
Level of significance: Criterion that
determines acceptance/rejection of H0
 Level of significance denoted as alpha (a)
 a = the probability of a type I error
 a can range between >0.0 – <1.0
 Typical values:

 10% chance of type I error
 0.05  5% chance of type I error
 0.01  1% chance of type I error
 0.10
Step 2: Significance Level
How to determine a?
 Exploratory research: Type I error is
acceptable therefore set higher

a

 0.05 – 0.10
When is type I error unacceptable?
Hypothesis Tests – A Step by Step
Process
Step 1: State the null hypothesis
 Step 2: Select level of significance
 Step 3: Sample data
 Step 4: Choose statistic
 Step 5: Calculate the statistic
 Step 6: Interpret the statistic

Step 3: Sample Data
Parametric statistics assume that data
were randomly sampled from population of
interest
 Generalization is limited to population that
was sampled

Hypothesis Tests – A Step by Step
Process
Step 1: State the null hypothesis
 Step 2: Select level of significance
 Step 3: Sample data
 Step 4: Choose statistic
 Step 5: Calculate the statistic
 Step 6: Interpret the statistic

Step 4: Choose the Statistic

Parametric or nonparametric?
 Scale

How many means are being compared?
 Two,

of measurement and distribution
three or more?
How are the means being compared?
 Between

How many independent variables (factors) are
being tested?
 Factorial

or within group?
design?
How many dependent variables are there?
 Multivariate
design?
Hypothesis Tests – A Step by Step
Process
Step 1: State the null hypothesis
 Step 2: Select level of significance
 Step 3: Sample data
 Step 4: Choose statistic
 Step 5: Calculate the statistic
 Step 6: Interpret the statistic

Step 5: Calculate the Statistic

Recall:
 exp. design  statistic
 The statistic tests the H0
 H0

A test statistic can be considered as a ratio
between:
 Between
variance (difference b/w means)
 Within variance (variability w/n means)
 Statistic = BV/WV

Large test statistics imply that:
 The
 The
difference between the means is relatively large
variance within the means is relatively small
Example: Researchers compare IQ scores between 6th grade
boys and girls. Results: Girls (150 +/- 50), boys (75 +/- 50)
Between Variance
Distribution
overlap?
Within Variance
0
50
150
200
Statistic = BV/WV
Statistic = Big / Big = small value
Statistic = Small / Small = small value
Statistic = Small / Big = small value
Statistic = Big / Small = Big value
Step 5: Calculate the Statistic


How does sample size affect the
statistic?
As sample size increases, the within
variance decreases  increases size of
test statistic
Hypothesis Tests – A Step by Step
Process
Step 1: State the null hypothesis
 Step 2: Select level of significance
 Step 3: Sample data
 Step 4: Choose statistic
 Step 5: Calculate the statistic
 Step 6: Interpret the statistic

Step 6: Interpret the Statistic
Calculation of the test statistic also yields
a p-value
 The p-value is the probability of a type I
error
 The p-value ranges from >0.0 – <1.0
 Recall alpha (a)
 a represents the maximum acceptable
probability of type I error therefore . . .

Step 6: Interpret the Statistic

If the p-value > a  accept the H0
 Probability
of type I error is higher than accepted
level
 Researcher is not “comfortable” stating that any
differences are real and not due to chance

If the p-value < a  reject the H0
 Probability
of type I error is lower than accepted
level
 Researcher is “comfortable” stating that any
differences are real and not due to chance
Statistical vs. Practical Significance

1.
2.
Distinction:
Statistical significance: There is an
acceptably low chance of a type I error
Practical significance: The actual
difference between the means are not
trivial in their practical applications