# If $p\text{-value}<\alpha$ does the observed test statistic always belongs to critical region?

Assume we carry out a hypothesis test at the 5% significance level. We have an observed test statistics $t$ with calculated p-value $0.03$. Does that imply that the observation has to lie in the critical region? I mean $3\%$ of the distribution is at least as extreme and the critical region is the most extreme 5% of the distribution, therefore $t$ must be contained in the critical region?

• That would be true if the t test is the appropriate test to use. Remember that the t test depends on normality. Mar 5, 2017 at 18:16
• what is observed test statistic ? What do I understand from critical region ?
– user10619
May 12, 2018 at 4:48
• Most often the value of the statistic and the p-value are related by a monotonous function. E.g. for a one sided t-test you have $$p=\int_{t_{observed}}^\infty f_\nu(t) dt$$ with $f_\nu(t)$ the t-distribution with $\nu$ degrees of freedom. So a smaller $p$ implies a larger $t_{observed}$. A $t_{observed}$ associated with a 0.03 p-value is in the 'region' of all t values that are larger than the t value associated with a 0.05 p-value. While this is all true Michael refers to the fact that a calculated p-value might not be realistic (underestimated) such that 0.03 may not be a critical value. May 12, 2018 at 6:21

If p-value<α does the observed test statistics always belongs to critical region?

Yes, that's right.

(It doesn't depend on whether the t-test is appropriate as suggested in comments -- the appropriateness of the assumptions doesn't come into this at all; this is a question of the decision you make when presented with a p-value. The appropriateness of the assumptions would matter when interpreting the p-value and it would matter in relation to the decision-process yielding the properties you desire, but none of that is at issue.)

I mean 3% of the distribution is at least as extreme and the critical region is the most extreme 5% of the distribution

This is correct. Anything up to (and including) 5% is at least as extreme as 5%.

• what is critical region in the present context ?
– user10619
May 12, 2018 at 4:41
• The term carries its usual meaning. It's the set of values of the test statistic that you would reject at a given significance level. For example, for a two-tailed t-test it is the largest values of $|t|$ that are in the critical region - specifically the fraction $\alpha$ (under the null distribution) of the largest values May 12, 2018 at 6:30
• This description has an orientation that whether t computed is significant or not significant. p-statistics connotes rejection rate of null hypothesis when it is true. If null distribution matches the observed distribution, and p is low, it means that model is fitted well and predicts well. it is not comparing t statistic with critical t- statistics to acertain a significant or nonsignificant effect. p-statistics evaluates goodness of fit - similar to F-statistic. A low rate of rejection of null hypothesis is indicative of a higher level of fitness of observed distribution.
– user10619
May 12, 2018 at 9:05
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://stats.stackexchange.com/questions/265584/if-p-text-value-alpha-does-the-observed-test-statistic-always-belongs-to-cr", "fetch_time": 1652835927000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2022-21/segments/1652662520936.24/warc/CC-MAIN-20220517225809-20220518015809-00274.warc.gz", "warc_record_offset": 606510578, "warc_record_length": 66105, "token_count": 766, "char_count": 3150, "score": 3.5625, "int_score": 4, "crawl": "CC-MAIN-2022-21", "snapshot_type": "latest", "language": "en", "language_score": 0.9298588633537292, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00014-of-00064.parquet"}