Give the recurrence formula for binomial distribution

1. Small sample:

The number of elements in the sample is less than $30$, then it is called a small sample.

2. $t$-distribution:

(i) For conducting $t$-tests the parent population(s) should be normal and the samples(s) should be small.

(ii) Sampling distribution of $t$ does not depend on population parameter. It depends on degrees of freedom $\left(n-1\right)$.

3. Applications of $t$-distribution:

(i) To test significance of a single population mean, when population variance is unknown.

(ii) To test the equality of two population means when population variances are equal and unknown.

(iii) To test the equality of two means paired $t$ -test, based on dependent samples.

4. Test statistic for test of hypotheses using $t$-distribution:

(i) For normal population mean when population variance is unknown is $T=\frac{\overline{X}-{\mu }_0}{\frac{S}{\sqrt{n}}}$ under $H_0$.

(ii) For Equality of means of two normal populations $T=\frac{(\overline{X}-\overline{y})}{S_p\sqrt{\frac{1}{m}+\frac{1}{n}}}$ under $H_0$, where $S_p=\frac{(m-1)s_x^2+(n-1)s_y^2}{m+n-2}$

(iii) For paired $t$-test $T=\frac{D}{{\displaystyle \frac{S}{\sqrt{n}}}}$ under $H_0$, where $S=\sqrt{\frac{{\sum }_{i=1}^n{\left(D_i-\overline{D}\right)}^2}{n-1}}$

5. Chi-square distribution $\left({\chi }^2\right)$ :

If $Z~N(0,1)$ then $Z^2~{\chi }^2$ with $1$ degree of freedom.

6. Applications of chi-square distribution:

(i) To test the variance of the normal population.

(ii) To test the independence of attributes.

(iii) To test the goodness of fit of a distribution.

7. Test statistic for test of hypotheses using chi-square distribution:

(i) For population variance of the normal population when population mean is assumed to be unknown is ${\chi }^2=\frac{(n-1)s^2}{{\sigma }_0^2}$ under $H_0$.

(ii) For independence of attributes is ${\chi }^2=\sum _{i=1}^m\sum _{j=1}^n\frac{\left({\left(O_{ij}-E_{ij}\right)}^2\right)}{E_{ij}}$ under $H_0$, where $E_{ij}=\frac{R_i\times C_j}{N}$.

(iii) For Goodness of fit ${\chi }^2=\sum _{i=1}^k\frac{{\left(O_i-E_j\right)}^2}{E_i}$ under $H_0$.

8. Expected frequency:

Expected frequency $=$ sample size $\times$ {probablity for the corresponding cell}.