Statistics: Inference, Hypothesis Testing, and Error Types - Prof. M. Seaman, Study notes of Applied Statistics

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Introduction toInference^ Probability, Risk, and^ Approximating Truth

Descriptive Statistics „^ Used to describe the characteristics ofempirical (sample) distributions „^ Provides research outcomes for a sample „^ Reveals relationships among variables inthe sample^ „^ Size of a correlation^ „^ Size of a difference^ „^ Specific variable effects

Types of Inferences „^ Population Inference^ „^ Infer from the sample statistic to a populationparameter^ „^ The sample statistic is compared to all possiblechance statistics „^ Causal Inference^ „^ Infer non-chance effects from one randomization^ „^ The sample outcome is compared to all possiblechance outcomes

Both types of inference rely on the observationof a subset taken from a set of all possibilities.

ALL POSSIBLE RANDOMIZATIONS Causal inference uses one randomization

ALL POSSIBLE SAMPLES Populationinferenceuses onesample

Hypothesis Testing „^ State an alternative hypothesis „^ State a null hypothesis „^ Test the null hypothesis^ „^ Estimate the size of the effect^ „^ Determine the conditional probability^ „^ Make a decision „^ Make inferences based on test results

The Null Hypothesis „^ Postulate a null hypothesis „^ Determine if the sample statistic is improbable,given the truth of the null hypothesis^ „^ If improbable, reject the null hypothesis^ „^ If plausible, retain the null hypothesis

: H h Ω^ = 0

Falsification^ „^ A theory must be stated so that it can befalsified by a finite set of observations^ „^ A scientific theory can only be falsified, neverproved correct^ „^ If a hypothesis does not receive support, thetheory may be incorrect in its present form^ „^ If a theory is repeatedly not supported, it shouldbe thrown out or revised^ „^ If a hypothesis is supported, it does not provethe theory correct

Characteristics of Hypotheses „^ Hypotheses can be made about anyparameter of interest „^ Hypotheses can be one- or two-sided^ „^ One-sided hypotheses are used to confirm atheoretical expectation^ „^ Two-sided hypotheses are used to explorepotential values of a parameter „^ A statistical hypothesis should parallel aresearch hypothesis

Type II Errors „^ Incorrectly retaining a null hypothesis is called aType II error „^ The consequence of a Type II error is a failure toreach a conclusion about an effect or parameter „^ The researcher can set the maximum probabilityof a Type II error (symbolized as

β) „^ The power of a hypothesis test is the probability ofrejecting a false null hypothesis (1 –

β)

Power 0.3 0.2 0.1^0 Statistic

REJECTIONREGION^ The Type I errorrate of a test isthe probability ofrejecting H

when (^0) His true 0 H0.5a 0.40.30.20.1 0 StatisticH^0

The power of atest is the actualprobability ofrejecting H^0

Researcher Controls „^ The maximum Type I error rate can be setby the researcher „^ Power can be set by the researcher „^ Sample size can be set by the researcher „^ The limitation is that only two of these threecontrols can be set at any one time

P-Values „^ A P-value is a measure of inconsistency with thenull hypothesis „^ P equals the probability of obtaining a value of astatistic that is at least as inconsistent with the nullhypothesis as the observed value of the statistic „^ Decisions about H

are based on the P-value and 0 the maximum Type I error rate^ „^ Reject H^ when P^0

≤ α „^ Retain Hwhen P >^0

α

Confidence Intervals^ A confidence interval consists of all values forthe parameter that are not rejected, given theobserved data, when we test all hypotheses.^ We would expect thatfour out of five 80%confidence intervalswill capture theparameter.^0

10 15

20

Cautions About Inference „^ Statistical inference is only as good as thedesign used to collect the data „^ Statistical significance does not implypractical significance „^ Failure to reject the null hypothesis doesnot prove that the null hypothesis is true „^ Conducting multiple hypotheses tests canlead to a compounding of errors