Analysis of sample size dynamics:

• Incorrect: Non-sampling error may increase with large scale surveys, but sampling error decreases.
• Incorrect: Standard error $\text{SE} = \sigma / \sqrt{n}$ decreases as $n$ increases.
• Correct: Sampling error is inversely proportional to the square root of the sample size ($n$). As sample size increases, the sample better approximates the population parameters, thereby steadily decreasing sampling error.
• Incorrect: Sample size has a direct mathematical impact on sampling error.

Chi-Square test of goodness of fit:

• Die would be deemed unfair only if calculated $\chi^2 \gt 11.07$.
• For a fair die thrown $60$ times, expected frequency for each of the $6$ faces is $E = \frac{60}{6} = 10$.
  Observed ($O$): $14, 7, 5, 8, 10, 16$.
  $\chi^2 = \sum \frac{(O - E)^2}{E} = \frac{(14-10)^2 + (7-10)^2 + (5-10)^2 + (8-10)^2 + (10-10)^2 + (16-10)^2}{10} = \frac{16 + 9 + 25 + 4 + 0 + 36}{10} = \frac{90}{10} = 9.0$.
  Since calculated $\chi^2 = 9.0 \lt 11.07$ (critical value at $\alpha = 0.05, df = 5$), we accept the null hypothesis that the die is fair.
• The test provides a clear statistical conclusion.
• Re-running is not required since sample size and expected counts are adequate.

Calculation of student's t-test statistic:

• $2.73$ is a typical critical t-value at df = 4.
• $0.97$ is an incorrect intermediate calculation.
• $3.30$ is incorrect.
• For sample $n = 8$: $\sum X_i = -4 - 3 - 3 + 0 + 3 + 3 + 4 + 4 = 4 \implies \bar{X} = 4/8 = 0.5$. Sum of squares $\sum X_i^2 = 16 + 9 + 9 + 0 + 9 + 9 + 16 + 16 = 84$. Sample variance $s^2 = \frac{\sum X_i^2 - n\bar{X}^2}{n - 1} = \frac{84 - 8(0.25)}{7} = \frac{82}{7} \approx 11.714$. Standard error $\text{SE} = \sqrt{s^2/n} = \sqrt{11.714/8} = \sqrt{1.464} \approx 1.210$. The test statistic is $t = \frac{\bar{X} - 0}{\text{SE}} = \frac{0.5}{1.210} \approx 0.413 \approx 0.41$.