fidence intervals:Confidence intervals: Because a Confidence intervalestimate of the population parameter contains thelikely values of that parameter, reject a claim that thepopulation parameter has a value that is not includedin the confidence intervalDecision Criterion
Statistical ErrorSometimes H0 will be rejected (based on large teststatistic & small p-value) even though H0 is really truei.e., if you had been able to measure the entirepopulation, not a sample, you would have foundno difference between and µ some value butbased on X you see a difference.The mistake of rejecting a true H0 will happen with frequency αSo, if H0 is true, it will be rejected ~5% of the time as α frequently = 0.05
00 20Population mean = 0Sample mean = 20Conclude based on sample mean that population mean ≠ 0, but it reallydoes (H0 true), therefore you have falsely rejected H0Type I Errorpopulation=“True”Sample=What you seeH0 : mean = 0
Statistical ErrorSometimes H0 will be accepted (based on small test statistic & largep-value) even though H0 is really falsei.e., if you had been able to measure the entirepopulation, not a sample, you would have founda difference between and µ some value- butbased on X you do not see a difference.The mistake of accepting a false H0 will happenwith frequency β
0Sample mean = 00 20Sample mean = 20Conclude based on sample mean that population mean = 0, but it reallydoes not (H0 really false), therefore you have falsely failed to reject H0Type II ErrorPopulation= “True”Sample= what you seeH0 : mean = 020
1. The treatments do not differ, and we correctly concludethat they do not differ.2. The treatments do not differ, but we conclude that theydo differ.3. The treatments differ, but we conclude that they do notdiffer.4. The treatments do differ, and we correctly conclude thatthey do differ.Four Possibilities in Testing Whetherthe Treatments Differ
Type I error∗ Concluded thatthere is differencewhile in realitythere is nodifference∗ α probabilityType II error∗Concluded thatthere is nodifference while inreality there is adifference∗β probability
Controlling Type I & Type II ErrorsαβPower (1 – β)Sample Size
∗ The power of the hypothesis test is the probability (1-(1- ββ))rejecting a false null hypothesis, which is computed byusing:∗ A particular significant level α∗ Sample size nn∗ A particular assumed value of the population parameterin the null hypothesis∗ A particular assumed value of the population parameterthat is alternative to the value in the null hypothesisPower of the test
Term Definitionsα = Probability of making a type I error= Probability of concluding the treatments differ when in realitythey do not differβ = Probability of making a type II error= Probability of concluding that the treatments do not differ whenin reality they do differPower = 1 - Probability of making a type II error= 1 - β= Probability of correctly concluding that the treatments differ= Probability of detecting a difference between the treatments ifthe treatments do in fact differ