Inference Concepts Hypothesis Testing

Confidence IntervalsSlide #2 Inference Sample Statistic Population Parameter Hypothesis/Significance Testing –assess evidence the data gives against a hypothesis about a parameter Confidence Regions –provide a range (region) believed to contain the parameter with a certain believability (confidence)

Inference ConceptsSlide #3 The Scientific Method Make Observation Make Predictions from Hypothesis Gather Observations / Experiment Compare Observations to Predictions Match, gain belief in hypothesis Don’t match, lose belief in hypothesis Construct Hypothesis

Inference ConceptsSlide #4 The Scientific Method Make Observation Make Predictions from Hypothesis Gather Observations / Experiment Compare Observations to Predictions Match, gain belief in hypothesis Don’t match, lose belief in hypothesis Construct Hypothesis Statistical hypothesis testing is at center Compares predictions to observations in the face of sampling variability Statistical hypothesis testing follows same logic Compares observations to predictions from a hypothesis

Inference ConceptsSlide #5 Two Main Hypothesis Types Research Hypothesis –a general statement of an effect Statistical Hypothesis (two types) –Alternative Hypotheses (H A ) a mathematical representation of the research hypothesis one of H A : parameter,  specific value –Null Hypotheses (H o ) the “no effect” or “no difference” situation always H o : parameter = specific value

What is the null and alternative hypotheses for these research hypotheses … “The mean density of Canada yew (Taxus canadensis) in areas not exposed to moose (Alces alces) on Isle Royale will be more than 1 stem per m 2 ” “The mean age of medical college students (Homo sapien) is less than 24 years” “The mean number of murders per burrough is less than 90” “The mean longevity of employees at the company is different than 10 years” Inference ConceptsSlide #6

Inference ConceptsSlide #7 An Example A research paper claims that the mean fetal heart rate is 137 bpm. A doctor feels that the mean rate is lower for women admitted to her clinic. What are the statistical hypotheses? H A :  < 137H O :  = 137 She will test her belief with … –a random sample of 100 patients –assuming  =10

Inference ConceptsSlide #8 The Null Hypothesis Assumed, initially, to be true –Used to predict what will be observed in a sample Thus, if H 0 :  =137 then one predicts that  x=137 IF there was no sampling variability, then what do you think about H 0 if  x=135 is observed

Inference ConceptsSlide #9 Does 135 support H 0 ? Does 134 support H 0 ? Does support H 0 ?      Sampling Variability Assume H 0 is true 137 SE – measure of sampling variability

Inference ConceptsSlide #10 Objectivity – p-value PR(observed statistic or value more extreme assuming H 0 is true) –shade to left if a “less than” H A –shade to right if a “greater than” H A –shade into both tails if a “not equals” H A One-Tailed Two-Tailed

Inference ConceptsSlide #11 An Example Suppose  x=135.9 was observed p-value = Pr(  x=135.9 or less, if  =137 )     p-value = distrib(135.9,mean=137,sd=1) 137 H A :  < 137

Inference ConceptsSlide #12 Objectivity – p-value Compare to rejection criterion –  –if p-value <  then reject H 0 –if p-value >  then Do Not Reject (DNR) H 0 Rejection criterion (  ) –“sets” cut-off value for determining support of H 0 –Set by researcher a priori –typical values are 0.10, 0.05, 0.01 PR(observed statistic or value more extreme assuming H 0 is true) Critical

Inference ConceptsSlide #13 An Example The doctor set  at 0.05 The p-value of is greater than  –Thus, DNR H 0 –Conclude that mean fetal heart rate for all of her patients is not less than 137 bpm The  x of likely occurred because of sampling variability and not a real difference from 137 bpm  135.9

For each situation below, write a definition of the p- value, compute the p-value, and make a decision –H A :  <80,  =40, n=50,  x=74,  =0.05 –H A :  >100,  =20, n=40,  x=105,  =0.05 –H A :  >100,  =20, n=80,  x=105,  =0.05 –H A :  ≠100,  =20, n=60,  x=103,  =0.10 –H A :  <100,  =20, n=40,  x=96,  =0.01 Inference ConceptsSlide #14

Inference ConceptsSlide #15 Summary Make statistical hypotheses from research hypothesis Use H 0 to make prediction (Assume H 0 is true) –this is why H 0 must be the “equals” situation Compare predicted statistic to observed statistic –calculate p-value Compare p-value to rejection criterion (  ) –if p-value >  then DNR H 0 conclude that H 0 could be correct –if p-value <  then reject H 0 conclude that H 0 is probably not correct Critical

Inference ConceptsSlide #16 if p-value >  then H 0 could be correct Recall that  x = for H 0 :  =137, p-value=0.1357, DNR H 0 for H 0 :  =137.1, p-value=0.1151, DNR H 0 for H 0 :  =137.2, p-value=0.0968, DNR H 0 for H 0 :  =137.3, p-value=0.0808, DNR H 0 There are always several other hypotheses that would also not be rejected. distrib(135.9,mean=137,sd=1) distrib(135.9,mean=137.1,sd=1) distrib(135.9,mean=137.2,sd=1) distrib(135.9,mean=137.3,sd=1)

Inference ConceptsSlide #17 DNR vs Accept The data do not contradict this H 0, but it is not fully known if this H 0 is true.

Inference ConceptsSlide #18      if p-value <  then H 0 is probably incorrect Even if H 0 is truly correct it is possible to observe a statistic in the tail, resulting in a p-value < , and a rejection of H 0.

Inference ConceptsSlide #19 Type II  Type I  Correct power Correct Decision Making Errors Set a priori by the researcher Can’t be known, because truth is not known

Inference ConceptsSlide #20 Effects on   is inversely related to  –i.e., “trading errors”  is inversely related to n –i.e., “more information means fewer errors”  is inversely related to difference between true and hypothesized value of parameter –i.e., “more obvious difference means fewer errors”