T-Test Calculator T-Test Calculator Compute the t‑statistic and p‑value for a one‑sample t-test using: $$ t = \frac{\bar{x} – \mu_0}{SD/\sqrt{n}}. $$ * Enter the sample mean (\(\bar{x}\)), hypothesized mean (\(\mu_0\)), sample standard deviation (SD), sample size (\(n\)), and select one‑tailed or two‑tailed test. Step 1: Enter Test Data Sample Mean, \(\bar{x}\): e.g., 100 Hypothesized Mean, […]
Read MoreKumaraswamy Distribution Variance Calculator Kumaraswamy Distribution Variance Calculator For the Kumaraswamy distribution with parameters \(a>0\) and \(b>0\), the \(k\)th moment is $$ E[X^k] = b\,\frac{\Gamma\Bigl(1+\frac{k}{a}\Bigr)\,\Gamma(b)}{\Gamma\Bigl(1+\frac{k}{a}+b\Bigr)}. $$ In particular, the mean is $$ \mu = b\,\frac{\Gamma\Bigl(1+\frac{1}{a}\Bigr)\,\Gamma(b)}{\Gamma\Bigl(1+\frac{1}{a}+b\Bigr)}, $$ and the variance is $$ \operatorname{Var}(X) = E[X^2] – \mu^2, $$ where $$ E[X^2] = b\,\frac{\Gamma\Bigl(1+\frac{2}{a}\Bigr)\,\Gamma(b)}{\Gamma\Bigl(1+\frac{2}{a}+b\Bigr)}. $$ * Enter […]
Read MoreKumaraswamy Distribution Mode Calculator Kumaraswamy Distribution Mode Calculator For the Kumaraswamy distribution with parameters \(a\) and \(b\), the PDF is given by: $$ f(x; a, b) = a\,b\, x^{a-1}\,(1 – x^a)^{b-1},\quad 0\le x\le 1. $$ * Enter parameters \(a>0\) and \(b>0\). – If \(a>1\) and \(b>1\), the interior mode is $$ x_{\mathrm{mode}} = \Bigl(\frac{a-1}{a\,b – […]
Read MoreKumaraswamy Distribution Median Calculator Kumaraswamy Distribution Median Calculator For the Kumaraswamy distribution with parameters \(a\) and \(b\), the CDF is given by: $$ F(x; a, b) = 1 – (1 – x^a)^b, \quad 0 \le x \le 1. $$ The median (the 50th percentile) satisfies \(F(x; a, b) = 0.5\) and can be computed as: […]
Read MoreKumaraswamy Distribution Mean Calculator Kumaraswamy Distribution Mean Calculator For the Kumaraswamy distribution with parameters \(a\) and \(b\), the mean is given by: $$ \mu = b\, B\Bigl(1+\frac{1}{a}, b\Bigr) = b\,\frac{\Gamma(1+1/a)\,\Gamma(b)}{\Gamma(1+1/a+b)}. $$ * Enter parameters \(a > 0\) and \(b > 0\). Step 1: Enter Parameters Parameter \(a\): e.g., 2 Parameter \(b\): e.g., 3 Calculate Mean […]
Read MoreKumaraswamy Distribution Inverse CDF Calculator Kumaraswamy Distribution Inverse CDF Calculator The Kumaraswamy distribution has CDF: $$ F(x; a, b) = 1 – (1 – x^a)^b, \quad 0 \le x \le 1. $$ Its inverse is given by: $$ F^{-1}(p; a, b) = \Bigl[ 1 – (1-p)^{\frac{1}{b}} \Bigr]^{\frac{1}{a}}, \quad 0 \le p \le 1. $$ * […]
Read MoreKumaraswamy Distribution CDF Calculator Kumaraswamy Distribution CDF Calculator For the Kumaraswamy distribution with parameters \(a\) and \(b\), the cumulative distribution function is given by: $$ F(x; a, b) = 1 – (1 – x^a)^b, \quad 0 \le x \le 1. $$ * Enter the value of \(x\) (between 0 and 1) and parameters \(a > […]
Read MoreKumaraswamy Distribution PDF Calculator Kumaraswamy Distribution PDF Calculator For the Kumaraswamy distribution with parameters \(a\) and \(b\), the probability density function is given by: $$ f(x; a, b) = a\, b\, x^{a-1}\,(1 – x^a)^{b-1}, \quad 0 \le x \le 1. $$ * Enter the value of \(x\) (between 0 and 1) and the parameters \(a […]
Read MoreStudentized Range Distribution Inverse CDF Calculator Studentized Range Distribution Inverse CDF Calculator This calculator finds the quantile \( q \) such that the Studentized Range CDF satisfies: $$ F(q; r, v)=p. $$ * Enter a probability \( p \) (0 < \( p \) < 1), number of groups \( r \) (\( r \ge […]
Read MoreStudentized Range Distribution CDF Calculator Studentized Range Distribution CDF Calculator This calculator computes the cumulative probability for a Studentized Range distribution. The PDF is given by: $$ f(q; r, v)=\frac{2\,\Gamma\Bigl(\frac{v+1}{2}\Bigr)}{\sqrt{\pi}\,\Gamma\Bigl(\frac{v}{2}\Bigr)}\,r\,q^{v-1}\int_{0}^{\infty} t^{v}e^{-t^2}\Bigl[\Phi\Bigl(\frac{q}{2}+\frac{t}{\sqrt{2}}\Bigr)-\Phi\Bigl(\frac{t}{\sqrt{2}}-\frac{q}{2}\Bigr)\Bigr]^{r-2}dt. $$ * Enter the Studentized Range value \( q \) (q ≥ 0), number of groups \( r \) (integer, \( r \ge 2 […]
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