Introduction to Hypothesis Introduction to Hypothesis TestingTesting

CJ 526 Statistical Analysis in CJ 526 Statistical Analysis in Criminal JusticeCriminal Justice

HypothesesHypotheses

A hypothesis is a prediction about the A hypothesis is a prediction about the outcome of a research studyoutcome of a research study

Hypothesis TestingHypothesis Testing

Hypothesis testing is an inferential Hypothesis testing is an inferential procedure that uses sample data to procedure that uses sample data to evaluate the credibility of a evaluate the credibility of a hypothesis about a populationhypothesis about a population

Overview of Hypothesis TestingOverview of Hypothesis Testing

1.1. State a hypothesis about a State a hypothesis about a populationpopulation

1.1. Usually in terms of the value of a Usually in terms of the value of a population parameterpopulation parameter

1.1. Typically the mean or the difference between Typically the mean or the difference between meansmeans

Overview of Hypothesis Testing Overview of Hypothesis Testing -- Continued-- Continued

If the data are consistent with the If the data are consistent with the hypothesis, conclude that the hypothesis, conclude that the hypothesis was reasonable, and fail hypothesis was reasonable, and fail to reject itto reject it

ExampleExample

Babies birth weight will not differ between Babies birth weight will not differ between smoking and non-smoking mothers (null)smoking and non-smoking mothers (null)

Babies born to women who smoke during Babies born to women who smoke during pregnancy will be more likely to be of low pregnancy will be more likely to be of low birth weight (alternative)birth weight (alternative)

Independent Variable:Independent Variable:• Smoking during pregnancySmoking during pregnancy

Dependent Variable:Dependent Variable:• Birth weightBirth weight

Example -- ContinuedExample -- Continued

1.1. Obtain a random sample of women who Obtain a random sample of women who are pregnant and smokeare pregnant and smoke

2.2. Obtain a random sample of non-smoking Obtain a random sample of non-smoking pregnant women, or compare to the pregnant women, or compare to the national averagenational average

3.3. Weigh the babies at birthWeigh the babies at birth4.4. Compare sample data to hypothesisCompare sample data to hypothesis5.5. Make decision:Make decision:

1.1. Reject the null hypothesisReject the null hypothesis2.2. Fail to reject the hypothesisFail to reject the hypothesis

Assumptions Behind Hypothesis Assumptions Behind Hypothesis TestingTesting

The effect of the Independent Variable The effect of the Independent Variable (treatment effect) is assumed to:(treatment effect) is assumed to:

Add (or subtract) a constant from every Add (or subtract) a constant from every individual’s scoreindividual’s score

The Logic of Hypothesis TestingThe Logic of Hypothesis Testing

1.1. Can’t prove hypothesisCan’t prove hypothesis1.1. Proof requires evidence for all casesProof requires evidence for all cases

Steps in Hypothesis TestingSteps in Hypothesis Testing

1.1. Determine the number of Determine the number of samples (groups, conditions)samples (groups, conditions)

1.1. OneOne

2.2. TwoTwo

3.3. k (three or more)k (three or more)

Steps in Hypothesis Testing -- Steps in Hypothesis Testing -- continuedcontinued

2.2. If there are two or more samples, If there are two or more samples, determine whether they are determine whether they are independent or dependentindependent or dependent

1.1. Same group (repeated-measures)Same group (repeated-measures)

2.2. Match on some other variable(s) known to Match on some other variable(s) known to influence DV (matched-subjects)influence DV (matched-subjects)

Steps in Hypothesis Testing -- Steps in Hypothesis Testing -- continuedcontinued

3.3. If there is one sample and the If there is one sample and the Dependent Variable is at the Dependent Variable is at the Interval or Ratio Level of Interval or Ratio Level of Measurement, is the standard Measurement, is the standard deviation of the population (deviation of the population (, , sigma) known:sigma) known:

1.1. If If is known, use a One-Sample z-Test is known, use a One-Sample z-Test

2.2. If If is unknown, use a One-Sample t-Test is unknown, use a One-Sample t-Test

Steps in Hypothesis Testing -- Steps in Hypothesis Testing -- continuedcontinued

4.4. Identify the independent variableIdentify the independent variable

5.5. Identify the dependent variable and Identify the dependent variable and its level of measurementits level of measurement

6.6. Identify the population to which Identify the population to which inferences will be madeinferences will be made

Steps in Hypothesis Testing -- Steps in Hypothesis Testing -- continuedcontinued

7.7. Determine the appropriate Determine the appropriate inferential statistical testinferential statistical test

1.1. Number of samplesNumber of samples

2.2. Nature of samples (if applicable)Nature of samples (if applicable)

3.3. Level of measurement of DVLevel of measurement of DV

8.8. State the null hypothesisState the null hypothesis

9.9. State the alternative hypothesisState the alternative hypothesis

Steps in Hypothesis Testing -- Steps in Hypothesis Testing -- continuedcontinued

10.10. State Decision Rule:State Decision Rule:1.1. If the p-value of the obtained test statistic If the p-value of the obtained test statistic

is less than .05, reject the Null Hypothesisis less than .05, reject the Null Hypothesis

11.11. Use Use SPSSSPSS to compute the obtained to compute the obtained test statistictest statistic

12.12. Make decisionMake decision

13.13. Interpret resultsInterpret results

Null HypothesisNull Hypothesis

The null hypothesis predicts that the The null hypothesis predicts that the Independent Variable (treatment) Independent Variable (treatment) will have no effect on the will have no effect on the Dependent Variable for the Dependent Variable for the populationpopulation

Alternative HypothesisAlternative Hypothesis

The alternative hypothesis predicts The alternative hypothesis predicts that the Independent Variable that the Independent Variable (treatment) will have an effect on (treatment) will have an effect on the Dependent Variable for the the Dependent Variable for the populationpopulation

Directional Alternative Directional Alternative HypothesesHypotheses

Researcher has reason to believe Researcher has reason to believe before conducting the test that a before conducting the test that a difference will lie in a specified difference will lie in a specified directiondirection

1.1. Prior researchPrior research

2.2. TheoryTheory

Non-directional Alternative Non-directional Alternative HypothesesHypotheses

Researcher has no reason to believe Researcher has no reason to believe that there will be a difference in a that there will be a difference in a specified directionspecified direction

There is insufficient research or There is insufficient research or information or theory to make a information or theory to make a specific predictionspecific prediction

Set the CriteriaSet the Criteria

Because of sampling error, there is Because of sampling error, there is likely to be a discrepancy between likely to be a discrepancy between the sample mean and the the sample mean and the population meanpopulation mean

At what point does the difference At what point does the difference become meaningful and not just a become meaningful and not just a matter of chance?matter of chance?

3. Collect Sample Data3. Collect Sample Data

Use the data to calculate the obtained Use the data to calculate the obtained test statistic, using the appropriate test statistic, using the appropriate statistical test, based on level of statistical test, based on level of measurement of the dependent measurement of the dependent variable, number of samples, variable, number of samples, whether the samples are whether the samples are independent or relatedindependent or related

4. Evaluate the Null Hypothesis4. Evaluate the Null Hypothesis

1.1. Reject the null hypothesisReject the null hypothesis1.1. If sample data is unlikely to have been If sample data is unlikely to have been

drawn from a population where the null drawn from a population where the null hypothesis is truehypothesis is true

2.2. If the p-value of the obtained test statistic If the p-value of the obtained test statistic is less than .05, meaning that the null is less than .05, meaning that the null hypothesis is rejected and there is less hypothesis is rejected and there is less than a 5% probability that this decision is than a 5% probability that this decision is incorrectincorrect

3.3. The alternative is accepted, that there is a The alternative is accepted, that there is a difference difference

OROR

Failure to Reject the Null Failure to Reject the Null HypothesisHypothesis

1.1. Either:Either:1.1. Treatment had an effect, could not Treatment had an effect, could not

demonstrate itdemonstrate it• oror

2.2. Treatment had no effectTreatment had no effect

Errors in Hypothesis TestingErrors in Hypothesis Testing

Actual StateActual State of Affairsof Affairs

BeliefBelief DecisionDecision HH00 is True is True HH00 is False is False

HH00 is False is False Reject HReject H00 Type I ErrorType I Error

False PositiveFalse Positive

Correct Correct RejectionRejection

1 - 1 - PowerPower

HH00 is True is True Fail to Reject Fail to Reject HH00

Correct Correct Failure to Failure to RejectReject

1 - 1 -

Type II ErrorType II Error

False False NegativeNegative

Type I ErrorType I Error

Committed when HCommitted when H00 is rejected as false is rejected as false although it is truealthough it is true

Type II ErrorType II Error

Committed when HCommitted when H00 is not rejected is not rejected although it is falsealthough it is false

Statistical PowerStatistical Power

Probability that the test will correctly Probability that the test will correctly reject a false null hypothesisreject a false null hypothesis

Power -- ContinuedPower -- Continued

When a treatment effect existsWhen a treatment effect exists1.1. A study may fail to discover it (Type II A study may fail to discover it (Type II

error, fail to reject a false null hypothesis)error, fail to reject a false null hypothesis)

2.2. A study may discover it (reject a false null A study may discover it (reject a false null hypothesis)hypothesis)

Power -- ContinuedPower -- Continued

Reducing alpha (.05 --> .01 --> .001)Reducing alpha (.05 --> .01 --> .001)1.1. Reduces powerReduces power2.2. Inverse relationship between Type I and Inverse relationship between Type I and

Type II errorsType II errors

Power -- ContinuedPower -- Continued

Some inferential statistical tests are Some inferential statistical tests are more powerfulmore powerful

Jury’s DecisionJury’s Decision

Did Not Commit CrimeDid Not Commit Crime Committed CrimeCommitted Crime

GuiltyGuilty Type I ErrorType I Error

Convict Innocent PersonConvict Innocent PersonCorrect VerdictCorrect Verdict

Convict Guilty Convict Guilty PersonPerson

Not GuiltyNot Guilty Correct AcquittalCorrect Acquittal

Fail to Convict Innocent Fail to Convict Innocent PersonPerson

Type II ErrorType II Error

Fail to Convict Fail to Convict Guilty PersonGuilty Person

Level of SignificanceLevel of Significance

Alpha: probability of committing a Alpha: probability of committing a Type I errorType I error

1.1. Reject HReject H00 although it is true although it is true2.2. Symbolized by Symbolized by