Education Research 250:205 Writing Chapter 3. Objectives Subjects Instrumentation Procedures Experimental Design Statistical Analysis Displaying data

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 populationWhat 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:

1. Shape

2. Central Tendency

3. Variability

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)

SEMMeasures how well statistic estimates

the parameterThe amount of sampling error that is

reasonable to expect by chance



SEM decreases when:Population decreasesSample size increases

Other properties:When n=1, SEM = population SD As SEM decreases the sampling distribution

“tightens”

SEM = /√n

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

Two main categories of inferential statistics

1. Parametric

2. Nonparametric

Introduction

Parametric or nonparametric? What is the scale of measurement?

Nominal or ordinal Nonparametric Interval or ratio Answer next question

Is the distribution normal?Yes ParametricNo Nonparametric





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 Components of a confidence interval:

1. The level of confidence-Chosen by researcher

-Typically 95%

-What does it mean?

2. The estimator (point estimate)

3. 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: The average $ spent was $56,789 (95% CI: $51,111

– 62,467) The average$ spent was $56,789 +/- 5,678 (95% CI)





Comparing Means Hypothesis Tests Compare two means

Compare a mean two a known value Compare means between groups Compare means within groups

Compare three or more means Compare means between groups Compare means within groups

Compare means as a function of two or more factors (independent variables) Factorial 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:Type I error Reject a true H0Type II error Accept a false H0

Step 1: Null HypothesisResearcher

Conclusion

Accept H0 Reject H0

Reality

About

Test

No real difference

exists

Correct

Conclusion

Type I error

Real difference exists

Type II error

Correct Conclusion

How does one control for Type I and II error?


Step 2: Significance Level Level of significance: Criterion that

determines acceptance/rejection of H0 Level of significance denoted as alpha () = the probability of a type I error can range between >0.0 – <1.0 Typical values:

0.10 10% chance of type I error0.05 5% chance of type I error0.01 1% chance of type I error

Step 2: Significance Level

How to determine ? Exploratory research: Type I error is

acceptable therefore set higher 0.05 – 0.10

When is type I error unacceptable?


Step 3: Sample Data

Parametric statistics assume that data were randomly sampled from population of interest

Generalization is limited to population that was sampled


Step 4: Choose the Statistic Parametric or nonparametric?

Scale of measurement and distribution How many means are being compared?

Two, three or more? How are the means being compared?

Between or within group? How many independent variables (factors) are

being tested? Factorial design?

How many dependent variables are there? Multivariate design?


Step 5: Calculate the Statistic

Recall: H0 exp. design statistic The statistic tests the 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 difference between the means is relatively large The 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)

0 20050 150

Between Variance

Within Variance

Distribution overlap?

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


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 () represents the maximum acceptable

probability of type I error therefore . . .

Step 6: Interpret the Statistic If the p-value > 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 < 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 Distinction:

1. Statistical significance: There is an acceptably low chance of a type I error

2. Practical significance: The actual difference between the means are not trivial in their practical applications

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Education Research 250:205 Writing Chapter 3. Objectives Subjects Instrumentation Procedures Experimental Design Statistical Analysis Displaying data