TN State Board 12th Statistics Question Paper 2023

The standard error of the sample mean is given by the formula $SE = \frac{\sigma}{\sqrt{n}}$. Option C matches this definition assuming standard notation.

In hypothesis testing, rejecting the null hypothesis $H_0$ when it is actually true is defined as a Type I error. Sampling error refers to the natural variation between samples, standard error is the measure of the dispersion of a statistic, and Type II error occurs when we fail to reject a false null hypothesis.

The degrees of freedom for a chi-square test in a contingency table of size $r \times c$ is calculated as $(r-1)(c-1)$. For a $5 \times 4$ contingency table, degrees of freedom $= (5-1)(4-1) = 4 \times 3 = 12$.

In two-way ANOVA classification, the Total Sum of Squares (TSS) is the sum of the sum of squares between rows (SSR), columns (SSC), and error (SSE). Therefore, $TSS = SSR + SSC + SSE$.

Spearman's rank correlation coefficient is given by $r_R = 1 - \frac{6 \Sigma D^2}{n(n^2-1)}$. If $\Sigma D^2 = 0$, then $r_R = 1 - \frac{6(0)}{n(n^2-1)} = 1 - 0 = 1$.