اختبار الفرضيات الإحصائي

اختبار الفرضيات الإحصائي Statistical hypothesis testing أو اختبار الفرضيات تعبر عن خوارزمية احصائية لإتخاذ قرار بشأن فرضية معينة تخص بيانات احصائية،أي تخص أحد الوسطاء مثل المتوسط أو التباين، أو مجتمع احصائي ما، القرار قد يكون بدعم الفرضية أو رفضها حسب مجال معين من الثقة تحدده طبيعة الدراسة و طبيعة البيانات الإحصائية، ويحدد الإختبار مدى انطباق البيانات المتوفرة مع الفرضية المدروسة مثل وجود علاقة بين خاصتين لأفراد المجتمع الإحصائي.

الإحصائيات الاختبارية الشائعة

See legend defining symbols at bottom of table. The statistics for some other tests have their own articles, including the Wald test and the likelihood ratio test.

الاسم	الصيغة	الافتراضات
One-sample z-test	$z=\frac{\overline{x}-\mu_0}{(\sigma / \sqrt{n})}$	(Normal distribution or n > 30) and σ known. (z is the distance from the mean in relation to the standard deviation of the mean). For non-normal distributions it is possible to calculate a minimum proportion of a population that falls within k standard deviations for any k (see: Chebyshev's inequality).
Two-sample z-test	$z=\frac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$	Normal distribution and independent observations and both (σ1 and σ2 known)
One-sample t-test	$t=\frac{\overline{x}-\mu_0} {( s / \sqrt{n} )} ,$ $df=n-1 \$	(Normal population or n < 30) and σ unknown
Paired t-test	$t=\frac{\overline{d}-d_0} { ( s_d / \sqrt{n} ) } ,$ $df=n-1 \$	(Normal population of differences or n < 30) and σ unknown
Two-sample pooled t-test	$t=\frac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{s_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}},$ $s_p^2=\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2},$ $df=n_1 + n_2 - 2 \$	(Normal populations or n1 + n2 > 40) and independent observations and σ1 = σ2 and (σ1 and σ2 unknown)
Two-sample unpooled t-test	$t=\frac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}},$ $df=\frac{(n_1 - 1)(n_2 - 1)}{(n_2 - 1)c^2 + (n_1 - 1)(1 - c)^2},$ $c=\frac{(\frac{s_1^2}{n_1})}{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})}$ or $df=\min\{n_1,n_2\} - 1\$	(Normal populations or n1 + n2 > 40) and independent observations and σ1 ≠ σ2 and (σ1 and σ2 unknown)
One-proportion z-test	$z=\frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$	n .p > 10 and n (1 − p) > 10 and it is a SRS (Simple Random Sample).
Two-proportion z-test, equal variances	$z=\frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$ $\hat{p}=\frac{x_1 + x_2}{n_1 + n_2}$	n1 p1 > 5 and n1(1 − p1) > 5 and n2 p2 > 5 and n2(1 − p2) > 5 and independent observations
Two-proportion z-test, unequal variances	$z=\frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}}$	n1 p1 > 5 and n1(1 − p1) > 5 and n2 p2 > 5 and n2(1 − p2) > 5 and independent observations
One-sample chi-square test	$\chi^2=\frac{(n-1)s^s}{\sigma^2}$	TODO
تعريف الرموز	$n$ = حجم العينة $\overline{x}$ = متوسط العينة $\mu_0$ = population mean $\sigma$ = population standard deviation $t$ = t statistic $df$ = degrees of freedom $n_1$ = sample 1 size $n_2$ = sample 2 size $s_1$ = sample 1 std. deviation $s_2$ = sample 2 std. deviation $\overline{d}$ = sample mean of differences $d_0$ = population mean difference $s_d$ = std. deviation of differences $p_1$ = proportion 1 $p_2$ = proportion 2 $\mu_1$ = population 1 mean $\mu_2$ = population 2 mean $\min\{n_1,n_2\}$ = minimum of n1 and n2 $x_1 = n_1 p_1$ $x_2 = n_2 p_2$

الأصول

Hypothesis testing is largely the product of Ronald Fisher, Jerzy Neyman, Karl Pearson and (son) Egon Pearson. Fisher was an agricultural statistician who emphasized rigorous experimental design and methods to extract a result from few samples assuming Gaussian distributions. Neyman (who teamed with the younger Pearson) emphasized mathematical rigor and methods to obtain more results from many samples and a wider range of distributions. Modern hypothesis testing is an (extended) hybrid of the Fisher vs Neyman/Pearson formulation, methods and terminology developed in the early 20th century.

انظر أيضاً

• Comparing means test decision tree
• Counternull
• Multiple comparisons
• Omnibus test
• Behrens–Fisher problem
• Bootstrapping (statistics)
• Checking if a coin is fair
• Falsifiability
• Fisher's method for combining independent tests of significance
• Null hypothesis
• P-value
• Statistical theory
• Statistical significance
• Type I error, Type II error
• Exact test

وصلات خارجية

• A Guide to Understanding Hypothesis Testing
• Hypothesis Testing: The Basics
• A good Introduction
• Bayesian critique of classical hypothesis testing
• Critique of classical hypothesis testing highlighting long-standing qualms of statisticians
• Dallal GE (2007) The Little Handbook of Statistical Practice (A good tutorial)
• References for arguments for and against hypothesis testing

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هذه المقالة عبارة عن بذرة تحتاج للنمو والتحسين؛ فساهم في إثرائها بالمشاركة في تحريرها.

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