 a score (z, tz, t or xx22))
∗ Critical region (Rejection region): Values of the teststatistic for which we reject the null in favor of thealternative hypothesisCritical Region, Significant level,Critical value and p-value
∗ Significant level (α ): the probability that the teststatistic will fall in the critical region when the nullhypothesis is actually true.Critical Region, Significant level,Critical value and p-value
∗ Critical value: is any value that separates the criticalregion from the values of the test statistic that do notlead to rejection of the null hypothesis .Critical Region, Significant level,Critical value and p-value
∗ Two tailed: the critical region is in the two extremeregions (tails) under the curveTwo-Tailed, Left Tailed, Right Tailed
∗ Left tailed: the critical region is in the extreme leftregion (tails) under the curveTwo-Tailed, Left Tailed, Right Tailed
∗ Right tailed: the critical region is in the extreme rightregion (tails) under the curveTwo-Tailed, Left Tailed, Right Tailed
∗ P-value (p-value or probability value: is theprobability of getting a value of the test statistic thatis at least as extreme as the one representing thesample data assuming the null hypothesis is true.∗ The null hypothesis is rejected if the p-value is verysmall such as 0.05 or less.Critical Region, Significant level,Critical value and p-value
Reject the null hypothesis (or other)Fail to reject the null hypothesisProve the null hypothesis to be trueAccept the null hypothesisSupport the null hypothesisStatisticallycorrectOk but misleading
∗ Traditional Method: Rejection of the null hypothesisif the statistic falls within the critical region Fail to reject the null hypothesis if the test statistic doesnot fall within the critical region∗ P – value methodP – value method: rejection H0 if p-value < α (where αis the significant level such as 0.05)Decision Criterion
∗ Another option:Another option: Instead of using a significant levelsuch as α = 0.05, simply identify the P value and leavethe decision to the reader∗ Confidence 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 inreali