Suppose you are preparing to conduct a test of

𝐻
0
:
𝜇
1
−
𝜇
2
=
0

𝐻
1
:
𝜇
1
−
𝜇
2

Question

Suppose you are preparing to conduct a test of

𝐻
0
:
𝜇
1
−
𝜇
2
=
0

𝐻
1
:
𝜇
1
−
𝜇
2
<
0

Your p-value is less than your significance level.

How would your test statistic relate to your critical value?

c08.p084.q019

BlackTom AI · Accepted Answer

We start by identifying the setup of the hypothesis test: a one-sided (left-tailed) test for the difference in means, H0: μ1 − μ2 = 0 versus H1: μ1 − μ2 < 0. When the p-value is less than the significance level, the observed test statistic falls into the rejection region of the null hypothesis. In a left-tailed test, the rejection region corresponds to very small values of the test statistic. Therefore, the observed test statistic would lie at a value more extreme in the left tail than the critical value, i.e., it would be smaller than the critical value. Concretely, if we denote the test statistic as TS and the critical value as CV, the relationship reflecting rejection at level α in this left-tailed framework is TS < CV.

To connect to the p-value concept: a small p-value indicates that the observed data are unlikely under H0, given the direction of the alternative hypothesis. Since H1(<) specifies negative differences, the data push the statistic toward more negative values, which is captured by TS falling below CV.

If, instead, the p-value were greater than α, we would fail to reject H0, and the test statistic would not lie in the left-tail rejection region (it would be TS ≥ CV in that setup). The key idea is that crossing into the rejection region for a left-tailed test corresponds to the test statistic being less than the critical value, aligning with TS < CV.

BTRY3010/BTRY5010/STSCI2200/STSCI5200 Lecture #27 (Pages 130 - 135 )

Suppose you are preparing to conduct a test of 𝐻 0 : 𝜇 1 − 𝜇 2 = 0 𝐻 1 : 𝜇 1 − 𝜇 2 < 0 Your p-value is less than your significance level. How would your test statistic relate to your critical value? c08.p084.q019

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