Paired-Samples T-Test Analysis and Statistical Significance

Standard Error Calculation

• Paired-samples t-test standard error ($S_M$) quantifies the expected variability between sample means:
  • Sample size: $N = 7$
  • Formula for standard error: $S_M = \frac{S}{\sqrt{N}}$
  • Calculation substitution: $\frac{0.45}{\sqrt{7}} = 0.17$
  • Resulting Standard Error ($S_M$): $0.17$

Critical Value Determination

• Parameters and criteria for establishing the rejection region boundaries:
  • Significance level: $\alpha = 0.05$
  • Test condition: Two-tailed test
  • Sample size: $N = 7$
  • Degrees of freedom formula: $df = N - 1$
  • Degrees of freedom calculation: $df = 7 - 1 = 6$
  • Critical values for $df = 6$ at $\alpha = 0.05$ (two-tailed): $-2.447$ and $2.447$

Test Statistic Calculation

• Computation of the test statistic ($t$):
  • Mean difference ($M_D$): $2.36$
  • Standard error of the difference ($S_M$): $0.17$
  • Formula: $t = \frac{M_D}{S_M}$
  • Calculation step: $t = \frac{2.36}{0.17} = 13.73$
  • Obtained test statistic: $t = 13.73$

Hypothesis Testing Decision and Statistical Significance

• Decision process for evaluating the null hypothesis:
  • Test statistic comparison: The calculated test statistic value ($13.73$) is compared against the critical value region defined by $\pm 2.447$.
  • Null hypothesis status: The null hypothesis is successfully rejected because the test statistic ($13.73$) falls beyond the critical threshold ($2.447$).
  • Conclusion on statistical significance: There is a statistically significant difference present in the data.

Confidence Interval Boundaries and Critical Values

• Test statistics for the 95% Confidence Interval (CI):
  • Study context: Paired-samples t-test with $df = 6$
  • Critical values applied: The 95% Confidence Interval uses the exact same critical boundaries as the two-tailed significance test, which are $-2.447$ and $2.447$ (also denoted as $2.477$).