Inferential StatisticsInferential Statistics- “data analysis techniques for determining how likely it is that - “data analysis techniques for determining how likely it is that results obtained from a sample or samples are the same results results obtained from a sample or samples are the same results that would have been obtained for the entire population” (p. that would have been obtained for the entire population” (p. 337)337)

-Techniques “used to make inferences about Techniques “used to make inferences about parametersparameters” (p. ” (p. 338)338)

- “ “using samples to make inferences about populations using samples to make inferences about populations produces only probability statements about the population” (p. produces only probability statements about the population” (p. 338)338)

-““analysis do not analysis do not proveprove that the results are true or false” (p. that the results are true or false” (p. 338)338)

Concepts underlying the application of Concepts underlying the application of Inferential Statistics:Inferential Statistics:- Standard Error: - Standard Error:

(S E x ) ) = (S E x ) ) =

SDSD

______________________________________________________

√ √ N - 1N - 1

- Samples can never truly reflect a Samples can never truly reflect a populationpopulation

- Variations among means of samples from Variations among means of samples from sample population is called sample population is called sampling sampling errorerror

- Sampling errors form a Sampling errors form a bell shaped bell shaped curvecurve

- Most of the sample means obtained will be Most of the sample means obtained will be close to the population meanclose to the population mean

- The The standard errorstandard error (SE x (SE x¯) tells us by ¯) tells us by how much we would expect our sample how much we would expect our sample mean to differ from the same populationmean to differ from the same population

The Null Hypothesis:The Null Hypothesis:

- A hypothesis stating that there is no A hypothesis stating that there is no relationship (or difference) between relationship (or difference) between variables and that any relationship found variables and that any relationship found will be a chance (not true) relationship, the will be a chance (not true) relationship, the result of sampling errorresult of sampling error

- Testing a Null hypothesis requires a Testing a Null hypothesis requires a test of test of significancesignificance and a and a selected probability selected probability levellevel that indicates how much risk you are that indicates how much risk you are willing to take that the decision you make is willing to take that the decision you make is wrongwrong

Tests of SignificanceTests of Significance

- Statistical tests used to determine whether or Statistical tests used to determine whether or not there is a significant difference between not there is a significant difference between or among two or more means at a selected or among two or more means at a selected probability levelprobability level

- Frequently used tests of significance are: Frequently used tests of significance are: t t test, analysis of variance, and chi squaretest, analysis of variance, and chi square

- Based on a test of significance the researcher Based on a test of significance the researcher will either reject or not reject the null will either reject or not reject the null hypothesishypothesis

Back toBack to Null Hypothesis Null Hypothesis

• Type I errorType I error: the researcher rejects : the researcher rejects a null hypothesis that is a null hypothesis that is really truereally true

• Type II errorType II error: the researcher fails to : the researcher fails to reject a hypothesis that is reject a hypothesis that is really really falsefalse

Probability level most Probability level most commonly used:commonly used:

- Is the alpha (Is the alpha (αα) where ) where αα = .05= .05

- If you select If you select αα as your probability as your probability level you have a 5% probability of level you have a 5% probability of making a Type I errormaking a Type I error

- The less chance of being wrong you The less chance of being wrong you want to take, the greater the want to take, the greater the difference of means must bedifference of means must be

Two-tailed and One-Tailed Two-tailed and One-Tailed teststests

- This is referring to the extreme ends of the This is referring to the extreme ends of the bell shaped curve that illustrates a normal bell shaped curve that illustrates a normal distributiondistribution

- A A two-tailed test two-tailed test allows for the allows for the possibility that a difference may occur in possibility that a difference may occur in either directioneither direction

- A A one-tailed test one-tailed test assumes that a assumes that a difference can occur in only one directiondifference can occur in only one direction

- Tests of significance Tests of significance are almost always are almost always two-tailedtwo-tailed

Degrees of FreedomDegrees of Freedom: :

- Dependent upon the number of Dependent upon the number of participantsparticipants and the number of and the number of groups groups

- Each test of significance has its own Each test of significance has its own formula for determining degrees of formula for determining degrees of freedomfreedom

- For Pearson r, the formula is NFor Pearson r, the formula is N= 2= 2

Types of Tests of Significance Types of Tests of Significance (choose the correct type)(choose the correct type)

(a)(a)Parametric testsParametric tests- used with - used with ratioratio and interval data, more and interval data, more powerfulpowerful, , more more often used, preferredoften used, preferred, but are , but are based on based on four major assumptions four major assumptions (p.348)(p.348)

(b)(b)Nonparametric tests Nonparametric tests used when the used when the data is data is nominal nominal or or ordinal, ordinal, when when parametric assumptions violated, or parametric assumptions violated, or when nature of distribution is unknown when nature of distribution is unknown

The t testThe t test::

- Used to determine whether two - Used to determine whether two means are significantly different at a means are significantly different at a selected probability level. There are selected probability level. There are two different types of t tests: two different types of t tests: t test t test for independent samples for independent samples and and t t test for non independent test for non independent samplessamples

- Independent samplesIndependent samples are two are two samples that are randomly formed samples that are randomly formed without any type of without any type of matchingmatching

- T test for independent samplesT test for independent samples is is a parametric test of significance used a parametric test of significance used to determine whether, at a selected to determine whether, at a selected probability level, a significant probability level, a significant difference exists between the means difference exists between the means of two independent samplesof two independent samples

--the t test the t test for for non independent non independent samples samples is used to determine is used to determine whether, at a selected probability whether, at a selected probability level, a significant difference exists level, a significant difference exists between the means of two matched, between the means of two matched, non independent, samplesnon independent, samples

- The formulae are: - The formulae are:

--you can also use you can also use SPSS 12. 0 SPSS 12. 0 to calculate to calculate t test t test for for independent and non independent samplesindependent and non independent samples

Sample Analysis of Variance Sample Analysis of Variance (ANOVA): (ANOVA): a parametric test of a parametric test of significance used to determine whether significance used to determine whether a significant difference exists between a significant difference exists between two or more means at a selected two or more means at a selected probability levelprobability level

-for a study involving -for a study involving three groupsthree groups ANOVA is the appropriate analysis ANOVA is the appropriate analysis techniquetechnique

An F ratio is computed: here An F ratio is computed: here are the twoare the two

- When the When the F ratioF ratio is significant and is significant and more than two means are involved, more than two means are involved, procedures called procedures called multicomparisonsmulticomparisons are used to determine which means are used to determine which means are significantly different from which are significantly different from which other meansother means

- The The ScheffScheffé testé test is appropriate for is appropriate for making any and all possible making any and all possible comparisons involving a set of means. comparisons involving a set of means. It involves calculation of an F ratio for It involves calculation of an F ratio for each mean comparison of interesteach mean comparison of interest

The ScheffThe Scheffé formulaé formula

- We can also use - We can also use SPSS 12.0 SPSS 12.0 to run to run multiple comparison tests to determine multiple comparison tests to determine which means are significantly different which means are significantly different from other meansfrom other means

Factorial Analysis of VarianceFactorial Analysis of Variance is a statistical technique thatis a statistical technique that::

- Allows the researcher to determine the Allows the researcher to determine the effect of the independent variable and the effect of the independent variable and the control variable on the dependent variable control variable on the dependent variable both separately and in combinationboth separately and in combination

- It is the appropriate statistical analysis if a It is the appropriate statistical analysis if a study is based on a factorial design and study is based on a factorial design and investigates two or more independent investigates two or more independent variables and the interactions between variables and the interactions between them- yeilds a separate F ratio for eachthem- yeilds a separate F ratio for each

Analysis of Covariance Analysis of Covariance (ANCOVA)(ANCOVA)

- A statistical method of equating - A statistical method of equating groups on one or more variables and groups on one or more variables and for increasing the power f a for increasing the power f a statistical test; adjusts scores on a statistical test; adjusts scores on a dependent variable for initial dependent variable for initial difference on other variablesdifference on other variables

Multiple regression equationMultiple regression equation::

- A prediction equation using two or - A prediction equation using two or more variables that individually more variables that individually predict a criterion in order to make a predict a criterion in order to make a more accurate predictionmore accurate prediction

Chi Square (X2): a Chi Square (X2): a nonparametric test of nonparametric test of significancesignificance- Appropriate when the data are in the Appropriate when the data are in the

form of form of frequencyfrequency count; compares count; compares proportions actually observed in a proportions actually observed in a study with expected proportions to study with expected proportions to see if they are significantly differentsee if they are significantly different

- There are two kinds of Chi square:There are two kinds of Chi square:

One Dimensional Chi square-One Dimensional Chi square- can be used to compare can be used to compare frequencies in different categoriesfrequencies in different categoriesTwo-Dimensional Chi square-Two-Dimensional Chi square- used when frequencies are used when frequencies are categorized along more than one categorized along more than one dimensiondimension- formulae:- formulae:

- Of course, you can also use - Of course, you can also use SPSS SPSS 12.0 12.0 to calculate to calculate Chi SquareChi Square