S.3.2 Hypothesis Testing (P | p value for t test
Inourexampleconcerningthemeangradepointaverage,supposethatourrandomsampleofn=15studentsmajoringinmathematicsyieldsateststatistict*equaling2.5.Sincen=15,ourteststatistict*hasn-1=14degreesoffreedom.Also,supposewesetoursignificancelevelαat0.05,sothatwehaveonlya5%chanceofmakingaTypeIerror.RightTailedTheP-valueforconductingtheright-tailedtestH0:μ=3versusHA:μ>3istheprobabilitythatwewouldobserveateststatisticgreaterthant*=2.5ifthepopulationmean(mu)reallywere3.Recallthatprobabilityequalstheareaun...
In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t* equaling 2.5. Since n = 15, our test statistic t* has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05, so that we have only a 5% chance of making a Type I error.
Right TailedThe P-value for conducting the right-tailed test H0 : μ = 3 versus HA : μ > 3 is the probability that we would observe a test statistic greater than t* = 2.5 if the population mean (mu) really were 3. Recall that probability equals the area under the probability curve. The P-value is therefore the area under a tn - 1 = t14 curve and to the right of the test statistic t* = 2.5. It can be shown using statistical software that the P-value is 0.0127. The graph depicts this visually.
The P-value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t* in the di...