(Individuals With Disabilities Act Transformation Over the Years) Discussion Forum Instructions: 1. You must post at least three times each week. 2. Your initial post is due Tuesday of each week and the following two post are due before Sunday.   3. All post must be on separate days of the week. 4. Post must be at least 150 words and cite all of your references even it its the book. Discussion Topic: Describe how the lives of students with disabilities from culturally and/or linguistically diverse backgrounds have changed since the advent of IDEA.  What do you feel are some things that can or should be implemented to better assist with students that have disabilities?  Tell me about these ideas and how would you integrate them? ANOVA ANOVA • Analysis of Variance • Statistical method to analyzes variances to determine if the means from more than two populations are the same • compare the between-sample-variation to the within-sample-variation • If the between-sample-variation is sufficiently large compared to the within-sample- variation it is likely that the population means are statistically different • Compares means (group differences) among levels of factors. No assumptions are made regarding how the factors are related • Residual related assumptions are the same as with simple regression • Explanatory variables can be qualitative or quantitative but are categorized for group investigations. These variables are often referred to as factors with levels (category levels) ANOVA Assumptions • Assume populations , from which the response values for the groups are drawn, are normally distributed • Assumes populations have equal variances • Can compare the ratio of smallest and largest sample standard deviations. Between .05 and 2 are typically not considered evidence of a violation assumption • Assumes the response data are independent • For large sample sizes, or for factor level sample sizes that are equal, the ANOVA test is robust to assumption violations of normality and unequal variances ANOVA and Variance Fixed or Random Factors • A factor is fixed if its levels are chosen before the ANOVA investigation begins • Difference in groups are only investigated for the specific pre-selected factors and levels • A factor is random if its levels are choosen randomly from the population before the ANOVA investigation begins Randomization • Assigning subjects to treatment groups or treatments to subjects randomly reduces the chance of bias selecting results ANOVA hypotheses statements One-way ANOVA One-Way ANOVA Hypotheses statements Test statistic = 𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑊𝑖𝑡ℎ𝑖𝑛 𝐺𝑟𝑜𝑢𝑝 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 Under the null hypothesis both the between and within group variances estimate the variance of the random error so the ratio is assumed to be close to 1. Null Hypothesis Alternate Hypothesis One-Way ANOVA One-Way ANOVA One-Way ANOVA Excel Output Treatme

1. (Individuals With Disabilities Act Transformation Over the
Years)
Discussion Forum Instructions:
1. You must post at least three times each week.
2. Your initial post is due Tuesday of each week and the
following two post are due before Sunday.
3. All post must be on separate days of the week.
4. Post must be at least 150 words and cite all of your
references even it its the book.
Discussion Topic:
Describe how the lives of students with disabilities from
culturally and/or linguistically diverse backgrounds have
changed since the advent of IDEA. What do you feel are some
things that can or should be implemented to better assist with
students that have disabilities? Tell me about these ideas and
how would you integrate them?
ANOVA
ANOVA
• Analysis of Variance
• Statistical method to analyzes variances to determine if the
means from more than
two populations are the same
• compare the between-sample-variation to the within-sample-
variation
• If the between-sample-variation is sufficiently large compared
to the within-sample-

2. variation it is likely that the population means are statistically
different
• Compares means (group differences) among levels of factors.
No
assumptions are made regarding how the factors are related
• Residual related assumptions are the same as with simple
regression
• Explanatory variables can be qualitative or quantitative but
are categorized
for group investigations. These variables are often referred to as
factors
with levels (category levels)
ANOVA Assumptions
• Assume populations , from which the response values for the
groups
are drawn, are normally distributed
• Assumes populations have equal variances
• Can compare the ratio of smallest and largest sample standard
deviations.
Between .05 and 2 are typically not considered evidence of a
violation
assumption
• Assumes the response data are independent
• For large sample sizes, or for factor level sample sizes that are

3. equal,
the ANOVA test is robust to assumption violations of normality
and
unequal variances
ANOVA and Variance
Fixed or Random Factors
• A factor is fixed if its levels are chosen before the ANOVA
investigation
begins
• Difference in groups are only investigated for the specific pre -
selected factors
and levels
• A factor is random if its levels are choosen randomly from the
population before the ANOVA investigation begins
Randomization
• Assigning subjects to treatment groups or treatments to
subjects
randomly reduces the chance of bias selecting results
ANOVA hypotheses statements

4. One-way ANOVA
One-Way ANOVA
Hypotheses statements
Test statistic
=
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Under the null hypothesis both the between and within group
variances estimate the
variance of the random error so the ratio is assumed to be close
to 1.
Null Hypothesis
Alternate Hypothesis
One-Way ANOVA
One-Way ANOVA
One-Way ANOVA Excel Output

5. Treatment
groups
Total 0.391495833 23 p-value is
found using
the F-statistic
and the F-
distribution.
The p-value for this
test is less than 0.05.
Reject the null
hypothesis and
conclude that at least
one treatment group
mean is statistically
different.
Multiple Comparison Tests
If the ANOVA test of the null hypothesis is rejected, then
conclude that not all the means are
equal but doesn’t suggest which means are statistically different
Multiple Comparison Test
One-Way ANOVA Example problem

6. One-Way ANOVA Example problem
One-Way ANOVA Example problem
Studentized Range q table
One-Way ANOVA Example problem
IntroDesire is to evalute the difference in effect between two or
more groupsThe groups are classified as levels of a factorwhen
there is only 1 factor, this is called a 1 way ANOVAsamples
are random and independent of each other so that group means
can be comparedANOVA compares means of groups by
analyzing the variation among and within groupstotal variation
is subdivided into variation due to differences between or
within groupsAssumptionsc groups are independently and
randomly selectedsample values in each group are from a
normal distributiongroups have equal variances (don't need to
worry if sample sizes in each group are the same)Null
hypothesis is no difference between meansunder the Null
hypothesis means of the c groups are assumed to be
equalcalculations sum the squared differences between group
means and the grand meancalculations also sum the squared
differences between individual group data and their group
meansnullalternateLittle Between group variancelots of between
group variance
Test StatisticThe ANOVA test statistic follows an F
distributionF statistic is a ratio of between group and within
group variance indiciesalpha - is the level of significance - used

7. to determine the critical F valuethe null hypotheis is rejected if
the value of the test statistic is greater than the critical F value
Critical F value
1-way ANOVA Summary TableStandard deviations for each
group can be obtained by taking the square root of the
variancesF>Fcrit then REJECT the null hypothesisp-value< .05
then REJECT the null hypothesisAt least one of the group
means is significantly different from one other (you don't know
which however)c-1 degrees of freedom for between groupn-c
degrees of freedom for within groupF = MS for between/ MS for
withinFcritical=3.6823203437MS = SS/dfWrite UpResults of
the ANOVA test shows that a statsitically significant difference
in group means (specify the groups) has been detected(F(2,15) =
7.15, p = .007)Note: Practically different and statistically
significantly different are not necessarily the same
Formulas for values in the ANOVA table
MSE
Example1Is there a meaningful difference in group means?You
needalpha0.05GroupAGroupBGroup CAnova: Single
Factor7111481412SUMMARY101410GroupsCou ntSumAverage
Variance121216GroupA5448.84.771013GroupB56112.23.2Grou
p C565135ANOVASource of VariationSSdfMSFP-valueF
critBetween
Groups49.7333333333224.86666666675.78294573640.0174334
863.8852938347Within
Groups51.6124.3Total101.333333333314
multiple comparison testThere are many different types of
multiple comparison tests (also called post hoc tests)Example:
Tukey HSD Multiple comparison testsimultaneous comparison
of means between all pairs of groups (there are other tests that
compare more than 2 means simultaeously)assumes equal
sample sizesSteps1compute the absolute mean differences
among all pairs of group meansthere will be c(c-1)/2 pairs of
meansuse the average column values in Excel SUmmary table to
obtian the group means2compute the HSD value The studentized
range (Q) is calculated for a particular alpha value for c and (n-

8. c) --- use the tablen is the group sample sizeMSE is found from
the ANOVA summary table (MS within groups
column)3compare each of the pairs of mean differences against
the HSD valueif the absolute difference in the sample means is
greater than the HSD value, then the means are significantly
different
Example2Tukey HSD Multiple ComparisonsWhat should the
minimal difference in group means be to indicate
significance?You needLevel of Significance (alpha)0.05c
(number of groups - k in this picture)3n (sample size)15n-
c12MSE (from ANOVA table)4.3Studentized Range (from
table)3.67GroupsCountSumAverageVarianceHSD (use
formula)1.9649642914GroupA5448.84.7GroupB56112.23.2Com
parisonsAbsolute DIfferenceHSDResults: Is HSD
smaller?Group C565135GroupA to
GroupB3.41.9649642914YessignificantGroupA to
GroupC4.21.9649642914YessignificantGroupB to
GroupC0.81.9649642914NoANOVASource of
VariationSSdfMSFP-valueF critBetween
Groups49.7333333333224.86666666675.78294573640.0174334
863.8852938347Within
Groups51.6124.3Total101.333333333314
Design of Experiments
Experimental Design
• Systematic process to layout, design, and investigate problems
in research and industry
• Starts with a screening process to identify factors (variables)
that may have an impact on

9. responses intended to be observed
• May consist of a number of iterative experiments to upgrade
and revise factors and details
understood regarding the factors
• Start by choosing factors that may affect results
• Factors that do not change during the experiments are referred
to as constants (or controlled)
• Optimization follows the screen process where the focus is on
establishing an approach
for predicting the response variable from the factors identified
during the screening
process
• Formulate hypothesis of the relation between factors and
predicted outcomes of the experiment
• Construct a statistical model that represents experiment results
• Testing follows optimization to ensure procedures are sound
and to evaluate the
robustness relative to fluctuations in factor conditions
Experimental Design
• The experimental design specifies the number and type of
experiments as well as the factors and their combinations used
for
each experiment, and how many times experiments will be run
(replicated) and in what order.
• What measurements to take (responses) is also specified
• Specify conditions and assumptions under which experiments
are run is
determined

10. • What levels (values) of each factor should be examined
• Resources and materials needed for the experiments are stated
• Run a series of simulations , or actual experiments, to refine
factor selection
and understanding of factor effects
Sections to include in Experimental Design
write up
• Title
• Hypothesis (questions to be investigated)
• Design type (simple, factorial, partial factorial)
• Factors and Levels of the factors
• Response variable and how it is measured
• Number of replications
• Constraints, assumptions, limitations of the experiment
Some frequent responses that are measured
• Mean
• Standard deviation
• Variance

11. • Standard error
Type of Experimental Designs
• Simple
• Start with a initial benchmark configuration and then vary one
factor level of a factor
at a time and then observe performance
• The number (n) of experiments to be conducted is
• Where ni is the number of levels for the ith factor. K is the
number of factors. For
example, in an experiment with 2 factors with 3 and 4 levels
respectively
n= 1 + (3-1) + (4-1) = 1 + 2 + 3 = 6 simple experiments are to
be conducted
Types of Experimental Designs
• Full factorial design
• For each successive experiment or simulation, investigate all
possible
combinations of all factor levels
• The number (n) of experiments to be conducted is
• For example, in an experiment with 2 factors with 3 and 4
levels respectively

12. n= 3x4 = 12 experiments are to be conducted
Type of Experimental Designs
• Fractional Factorial Designs
• Uses a subset of factors and factor levels in the experiments
• Experimental groupings
• Conducting a full factorial design investigation may not be
warranted or
feasible
• Number of experiments to conduct is determined by
multiplying together the
subset of factor levels of the select factors that will be
investigated
Power of a design
• Used to address issues of experimental accuracy
• Determine sample size needed to measure the response at a
desired level of
accuracy
Experimental Design common errors
• Ignoring causes of experimental error that may impact results

13. • All important factors, that can impact results, have not been
identified
• When varying several factors, have a lack of clarity to what
the actual
effect is
• Using multiple simple designs as opposed to a factorial design
• Not examining the impact of interactions between factors … if
factor
level effects are being influenced by the presence of other
factor
levels