Short Answer 
Clarke’s confidence interval for the probability of rolling a 6 suggests that the die may not be biased, as it includes the fair value of 1/6, indicating weak evidence of bias. In contrast, Aurelia’s significance test shows a p-value of 0.033, leading to the rejection of the null hypothesis and suggesting that the die does favor the number 6. Additionally, Joachim’s one-sided interval indicates that values for rolling a 6 are greater than 0.23, further supporting the conclusion of bias in favor of rolling a 6.
Step 1: Understanding Clarke’s Confidence Interval
Clarke’s confidence interval for the proportion of rolling a 6 with the manipulated die is calculated as 0.215 ¬¨¬± 0.057. This interval ranges from 0.158 to 0.272 and includes the value of 1/6 (0.167), which indicates that the die may not be favoring the number 6 more than a fair die. Since this value is part of the interval, it implies there isn’t strong statistical evidence to claim the die is biased toward the number 6.
Step 2: Analyzing Aurelia’s Significance Test
Aurelia’s significance test results provide a complementary perspective, where the p-value is recorded at 0.033. This p-value is less than the significance level of 0.05, leading to the rejection of the null hypothesis. Hence, this suggests there exists statistically significant evidence that the number 6 appears more frequently on the baked die compared to a standard die.
Step 3: Exploring Joachim’s One-Sided Confidence Interval
Joachim’s analysis utilizes a one-sided confidence interval, with a critical value Z* of 1.645. This results in a lower limit (L) of approximately 0.23, indicating that all plausible values of p for rolling a 6 are greater than this threshold. Thus, Joachim’s findings support Aurelia’s conclusions, suggesting the number 6 is indeed rolled more often on the manipulated die.