Understanding Statistical Errors in the King Dushyant Narrative

This question asks us to identify the type of statistical error King Dushyant committed based on the narrative from "Abhigyan Shankuntalam". The core issue is King Dushyant's failure to recognize Shakuntala because she lacked the royal ring, which was the expected proof of her identity as his wife.

What are Statistical Errors?

In statistics, particularly in hypothesis testing, we make decisions based on data. There's always a possibility of making an incorrect decision. These incorrect decisions are classified into two main types of errors:

• Type I Error: This occurs when we reject a null hypothesis ($H_0$) that is actually true. It's often called a "false positive". We conclude there is an effect or relationship when there isn't one in reality. The probability of a Type I error is denoted by $\alpha$.
• Type II Error: This occurs when we fail to reject a null hypothesis ($H_0$) that is actually false. It's often called a "false negative". We fail to detect an effect or relationship that actually exists. The probability of a Type II error is denoted by $\beta$.

Let's look at the possible outcomes of a hypothesis test:

Applying Statistical Error Concepts to the Narrative

Let's frame the situation in the narrative in terms of hypothesis testing:

• Null Hypothesis ($H_0$): Shakuntala is King Dushyant's wife (the marriage is true).
• Alternative Hypothesis ($H_1$): Shakuntala is not King Dushyant's wife (the marriage is not true or not recognized).

King Dushyant used the presence of the royal ring as his test or criterion. His decision rule was essentially:

• If ring is present, conclude $H_0$ is true (recognize her as wife).
• If ring is absent, conclude $H_0$ is false (do not recognize her as wife).

Now, let's consider the reality and the King's decision:

• Actual Situation: Shakuntala *was* truly King Dushyant's wife. So, the Null Hypothesis ($H_0$) was True.
• King's Decision: Shakuntala had lost the ring, so the test failed (ring absent). Based on his criterion, the King failed to recognize her, which means he effectively Rejected the Null Hypothesis ($H_0$).

Comparing the Actual Situation ($H_0$ is True) with the Decision (Reject $H_0$), we see that the King rejected a true null hypothesis.

This matches the definition of a Type I error.

If it had been a Type II error, it would mean Shakuntala was *not* his wife ($H_0$ is false), but he incorrectly recognized her anyway (failed to reject $H_0$). That is not what happened in the story.

Conclusion on the Statistical Error

In the narrative, King Dushyant's condition for recognition was the presence of the royal ring. When the ring was absent, he concluded she was not his wife, despite the fact that she truly was. Rejecting the true null hypothesis (that she is his wife) because the evidence (the ring) was missing constitutes a Type I error.

Revision Table: Understanding Errors

Additional Information: Hypothesis Testing Basics

Hypothesis testing is a fundamental concept in statistics used to make inferences about a population based on sample data. It involves setting up two competing hypotheses:

• Null Hypothesis ($H_0$): This is the statement being tested. It usually represents a status quo, a claim of no effect, or no difference. We assume $H_0$ is true until proven otherwise.
• Alternative Hypothesis ($H_1$ or $H_a$): This is the statement we want to conclude is true if the data provide sufficient evidence to reject the null hypothesis. It typically represents a claim of an effect or difference.

The process involves collecting data, performing a statistical test, and calculating a p-value. The p-value is the probability of observing data as extreme as, or more extreme than, the sample data, assuming the null hypothesis is true. Based on the p-value and a chosen significance level ($\alpha$), we decide whether to reject or fail to reject the null hypothesis.

• If p-value $\le \alpha$, we reject $H_0$.
• If p-value > $\alpha$, we fail to reject $H_0$.

The significance level ($\alpha$) is the maximum acceptable probability of committing a Type I error. Common values for $\alpha$ are 0.05 (5%) or 0.01 (1%). There is a trade-off between Type I and Type II errors; reducing the probability of one type of error often increases the probability of the other, for a fixed sample size.