Decision rule in statistical hypothesis testing:

• The critical region (rejection region) is the set of values of the test statistic for which the null hypothesis ($H_0$) is rejected at a chosen significance level ($\alpha$).
• If the computed test statistic (e.g. $t_{\text{cal}}$) falls inside the critical region ($|t_{\text{cal}}| \gt t_{\text{crit}}$), the observed difference is statistically significant, leading directly to the rejection of the null hypothesis ($H_0$).
  
  Option analysis:
• Reject $H_0$ is the fundamental decision rule in statistics (Key 1).
• Occurs when the statistic is outside the critical region.
• and (D) Option 4: Tests explicitly evaluate $H_0$, not directly rejecting $H_1$.

Effect of sample size on confidence interval width:

• The width of a confidence interval for a population mean is given by: $\text{Width} = 2 \times z^* \times \frac{\sigma}{\sqrt{n}}$, where $n$ is the sample size and $\frac{\sigma}{\sqrt{n}}$ is the standard error.
• As the sample size ($n$) increases, the standard error decreases inversely with $\sqrt{n}$.
• Consequently, the margin of error decreases, making the confidence interval narrower (decreases in size), providing a more precise estimate.
  
  Option analysis:
• Incorrect, width decreases.
• The confidence interval decreases in size (width shrinks) (Key 2).
• Incorrect, strongly dependent on $n$.
• Incorrect, it approaches zero only as $n \to \infty$, but does not become zero for any finite $n$.

Statistical notations:

• $R^2$ (or $r^2$): Coefficient of determination, measuring the proportion of variance in the dependent variable explained by the independent variable(s) in a regression model.
• $r$ (or $R$): Coefficient of correlation (Pearson correlation coefficient).
• $CV$: Coefficient of variation, calculated as $\frac{\sigma}{\mu} \times 100\%$.
• $b$ (or $\beta$): Coefficient of regression (slope of regression line).
  
  Option analysis:
• Noted as $r$.
• The coefficient of determination is universally denoted by $R^2$ (Key 2).
• Noted as $CV$.
• Noted as $b$.